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    <p class=3Dmsoaddress style=3D'mso-pagination:none'><span lang=3DEN-US
    style=3D'mso-ansi-language:EN-US'>&nbsp;<o:p></o:p></span></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  <![endif]></v:textbox>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout;positi=
on:
relative;z-index:27'><span style=3D'position:absolute;left:-25px;top:-77px;
width:546px;height:125px'><img width=3D546 height=3D125
src=3D"relativistic_files/image001.gif" v:shapes=3D"_x0000_s1107 _x0000_s11=
08"></span></span><![endif]><b
style=3D'mso-bidi-font-weight:normal'><span style=3D'font-size:16.0pt'><o:p=
>&nbsp;</o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:16.0pt'><o:p>&nbsp;</o:p></span></b></p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:16.0pt'>Title: Solution of Relativistic Schrodinger&#821=
7;s
Equation in Quantum Mechanics.<o:p></o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>By Professor Kwaku Boateng</span></b><span style=3D'font-size:14.0pt=
'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Tel<=
span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>00233 243 241 943<span
style=3D'mso-spacerun:yes'>&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:14.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;
</span></span></b><span style=3D'font-size:14.0pt'>00233 20 274 8355<b
style=3D'mso-bidi-font-weight:normal'><o:p></o:p></b></span></p>

<p class=3DMsoNormal><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>Email:
<a href=3D"mailto:info@newquantummechanics.com">info@newquantummechanics.co=
m</a></p>

<p class=3DMsoNormal style=3D'text-align:justify'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'text-align:justify'>Schrodinger gave the equa=
tion
below Fig 1.0</p>

<p class=3DMsoNormal style=3D'text-align:justify'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'text-align:justify'><o:p>&nbsp;</o:p></p>

<table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0 wi=
dth=3D461
 style=3D'width:345.6pt;border-collapse:collapse;border:none;mso-border-alt=
:solid windowtext .5pt;
 mso-yfti-tbllook:1184;mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-border-insid=
eh:
 .5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'>
  <td width=3D461 valign=3Dtop style=3D'width:345.6pt;border:solid windowte=
xt 1.0pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
  style=3D'font-size:16.0pt;mso-no-proof:yes'><!--[if gte vml 1]><v:shapety=
pe
   id=3D"_x0000_t75" coordsize=3D"21600,21600" o:spt=3D"75" o:preferrelativ=
e=3D"t"
   path=3D"m@4@5l@4@11@9@11@9@5xe" filled=3D"f" stroked=3D"f">
   <v:stroke joinstyle=3D"miter"/>
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/>
   <o:lock v:ext=3D"edit" aspectratio=3D"t"/>
  </v:shapetype><v:shape id=3D"_x0000_i1025" type=3D"#_x0000_t75" style=3D'=
width:370.5pt;
   height:119.25pt;visibility:visible'>
   <v:imagedata src=3D"relativistic_files/image002.jpg" o:href=3D"mhtml:fil=
e://F:\universal_websites\word_pages_=3D20062008\wordintroductiontoquantumm=
echanics.mht!http://www.physlink.com/Education/AskExperts/Images/ae329a.jpg=
"/>
  </v:shape><![endif]--><![if !vml]><img border=3D0 width=3D494 height=3D159
  src=3D"relativistic_files/image003.jpg" v:shapes=3D"_x0000_i1025"><![endi=
f]></span></b><b
  style=3D'mso-bidi-font-weight:normal'><span style=3D'font-size:20.0pt'><o=
:p></o:p></span></b></p>
  </td>
 </tr>
</table>

<p class=3DMsoNormal style=3D'text-align:justify'><b style=3D'mso-bidi-font=
-weight:
normal'><span style=3D'font-size:20.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span></span></b></p>

<p class=3DMsoNormal style=3D'text-align:justify'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.0</p>

<p><span class=3Dgoogqs-tidbit1><span lang=3DEN-US>In order to extend
Schr&ouml;dinger's formalism to include relativity, the physical picture mu=
st
be altered. The Klein-Gorden equation </span></span><span lang=3DEN-US>uses=
 the
relativistic mass-energy relation:</span></p>

<p class=3DMsoNormal style=3D'margin-left:36.0pt'><span style=3D'mso-no-pro=
of:yes'><!--[if gte vml 1]><v:shape
 id=3D"Picture_x0020_1" o:spid=3D"_x0000_i1026" type=3D"#_x0000_t75" alt=3D=
"E^2 =3D p^2c^2 + m^2c^4\,"
 style=3D'width:111pt;height:16.5pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image004.gif" o:title=3D","/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D148 height=3D22
src=3D"relativistic_files/image004.gif" alt=3D"E^2 =3D p^2c^2 + m^2c^4\," v=
:shapes=3D"Picture_x0020_1"><![endif]></span></p>

<p><span lang=3DEN-US>to produce the differential equation</span></p>

<table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0 wi=
dth=3D319
 style=3D'width:239.3pt;border-collapse:collapse;border:none;mso-border-alt=
:solid windowtext .5pt;
 mso-yfti-tbllook:1184;mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-border-insid=
eh:
 .5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'>
  <td width=3D319 valign=3Dtop style=3D'width:239.3pt;border:solid windowte=
xt 1.0pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'mso-no-proof:yes'><!--[if gte vml 1]>=
<v:shape
   id=3D"Picture_x0020_2" o:spid=3D"_x0000_i1027" type=3D"#_x0000_t75" alt=
=3D"- \frac{1}{c^2}{\partial^2 \over \partial t^2}\psi =3D  - \nabla^2 \psi=
 + \frac{m^2c^2}{\hbar^2} \psi\,"
   style=3D'width:180pt;height:33pt;visibility:visible'>
   <v:imagedata src=3D"relativistic_files/image005.gif" o:title=3D","/>
  </v:shape><![endif]--><![if !vml]><img border=3D0 width=3D240 height=3D44
  src=3D"relativistic_files/image005.gif"
  alt=3D"- \frac{1}{c^2}{\partial^2 \over \partial t^2}\psi =3D  - \nabla^2=
 \psi + \frac{m^2c^2}{\hbar^2} \psi\,"
  v:shapes=3D"Picture_x0020_2"><![endif]></span><b style=3D'mso-bidi-font-w=
eight:
  normal'><span style=3D'font-size:20.0pt'><o:p></o:p></span></b></p>
  </td>
 </tr>
</table>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>which is relativistically invariant.</p>

<p><span lang=3DEN-US>As before we replace the Greek letter by Y(x,t).</spa=
n></p>

<p><span lang=3DEN-US>Also assuming that x, and t are independent and also =
Y(x,t)
<span style=3D'mso-spacerun:yes'>&nbsp;</span>are seperable</span></p>

<p><span lang=3DEN-US>Y(x,t)=3DX(x)T(t)</span></p>

<p><span lang=3DEN-US>Where X is a space variable and T is time<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span></span></p>

<p><span lang=3DEN-US>Also denoting </span></p>

<p><span lang=3DEN-US style=3D'font-size:16.0pt'>X<sub>xx</sub></span><span
lang=3DEN-US> as derivatives with respect to x</span></p>

<p><span lang=3DEN-US style=3D'font-size:16.0pt'>T<sub>tt</sub></span><span
lang=3DEN-US><span style=3D'mso-spacerun:yes'>&nbsp; </span>as derivatives =
with
respect to t (time)</span></p>

<p class=3DMsoNormal>The equation above reduces to two equations given as</=
p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'mso-spacerun:yes'>&nbsp;</span><span
style=3D'font-size:16.0pt'>X<sub>xx<span style=3D'mso-spacerun:yes'>&nbsp; =
</span>+
</sub><span style=3D'mso-spacerun:yes'>&nbsp;</span>EX=3D0</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>T<sub>tt + </sub>k<su=
p>2</sup>
T<sub>tt<span style=3D'mso-spacerun:yes'>&nbsp; </span></sub>=3D0</span><su=
b><span
style=3D'font-size:20.0pt'><o:p></o:p></span></sub></p>

<p class=3DMsoNormal>Assuming V is small or zero and <span style=3D'font-si=
ze:16.0pt'>k<sup>2</sup></span>=3D<span
style=3D'font-size:16.0pt'> C<sup>2</sup></span> (E+ <span style=3D'font-si=
ze:16.0pt'>m<sub>e</sub><sup>2</sup>
C<sup>2/</sup> h<sup>2 </sup></span>)</p>

<p class=3DMsoNormal>That is </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>k=
=3D+or
&#8211; square root(<span style=3D'font-size:16.0pt'>C<sup>2</sup></span> (=
E+ <span
style=3D'font-size:16.0pt'>m<sub>e</sub><sup>2</sup> C<sup>2/</sup> h<sup>2=
 </sup></span>)</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>We also assume that the final solution will pass throu=
gh
points </p>

<p class=3DMsoNormal>Which are multiples of wavelengths of the waves. At any
point the time is distance divided by the speed of light.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>We get solutions of the form</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>T(t)=3DA. sin(kt)<o:p=
></o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>The X<span style=3D'mso-spacerun:yes'>&nbsp; </span>pa=
rt is
solved as follows.</p>

<p class=3DMsoNormal>This equation has solution under three main mathematic=
al
conditions which are stated here without proof:</p>

<p class=3DMsoNormal style=3D'margin-left:18.0pt;text-indent:-18.0pt;mso-li=
st:l0 level1 lfo2'><![if !supportLists]><span
style=3D'mso-list:Ignore'>(1)<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;&nbsp;
</span></span><![endif]>Dirichlet&#8217;s boundary conditions</p>

<p class=3DMsoNormal style=3D'margin-left:18.0pt;text-indent:-18.0pt;mso-li=
st:l0 level1 lfo2'><![if !supportLists]><span
style=3D'mso-list:Ignore'>(2)<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;&nbsp;
</span></span><![endif]>Neumann initial conditions</p>

<p class=3DMsoNormal style=3D'margin-left:18.0pt;text-indent:-18.0pt;mso-li=
st:l0 level1 lfo2'><![if !supportLists]><span
style=3D'mso-list:Ignore'>(3)<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;&nbsp;
</span></span><![endif]>Hermitian condition</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D</p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Case 1<o:p></o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Table 1<o:p></o:p></s=
pan></p>

<table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0
 style=3D'border-collapse:collapse;border:none;mso-border-alt:solid windowt=
ext .5pt;
 mso-yfti-tbllook:480;mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-border-inside=
h:
 .5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Function <o:p></o:p=
></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Dirichlet&#8217;s B=
oundary Conditions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Neumann Initial Con=
ditions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Hermitian Condition=
<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:1'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(0)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:2'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(a)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:3'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(0)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>d<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>D<o:p></o:p></span>=
</p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(a)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
 </tr>
</table>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>From table 1 we see t=
hat the
Hermitian conditions for the existence of solution with real values of ener=
gy
as<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(0)=3DX(a)=3D0 <o:p>=
</o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(a)=3D0<o:p><=
/o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>This means that we se=
ek
solutions of the equation in the form <span
style=3D'mso-spacerun:yes'>&nbsp;</span>as seen in Fig1.1.1; that is the Ei=
gen
function passes through x=3D0 and x=3Da where a=3D4 in this case.<o:p></o:p=
></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1041" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:1;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:1'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1041" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1028" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image006.png" o:title=3D"graph =
fig 1 or 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image007.gif" v:shapes=3D"_x0000_i1028"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.1.1<o:p></o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:20.0pt'><o:p>&nbsp;</o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:20.0pt'>Solution of equation<o:p></o:p></span></b></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>We
seek solution under the boundary condition<o:p></o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>S<span style=3D'font-family:Tahoma'>olution of the equation is given=
 as </span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>X
=3D Asin</span><span style=3D'font-size:22.0pt'>(&#955;</span><span
style=3D'font-family:Tahoma'> X) + BCos(</span><span style=3D'font-size:22.=
0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> x)<span style=3D'mso-spacerun:yes'>&nbsp;
</span>--------------- <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>(1)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>X<su=
p>1</sup>=3D
B</span><span style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-f=
amily:
Tahoma'> sin(</span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> x)<span style=3D'mso-spacerun:yes'>&nbsp;&nbs=
p;&nbsp;
</span>+ A </span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> Cos(</span><span style=3D'font-size:22.0pt'>&=
#955;</span><span
style=3D'font-family:Tahoma'> x) ------ (2)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Putt=
ing in
the value x=3D0 in line 1 gives <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Also=
 by
substituting x=3D1 in X(1)<span style=3D'mso-spacerun:yes'>&nbsp; </span>=
=3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Give=
s <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>X(1)
=3D<span style=3D'mso-spacerun:yes'>&nbsp; </span>Asin</span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
 =3D0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Hence<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:2=
2.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> =3D</span><span style=3D'font-size:16.0pt'> n=
&#960;</span><span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p=
></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>X<sub>n</sub> =
=3D A<sub>n</sub>
*sin</span><span style=3D'font-size:22.0pt'>(&#955;</span><span style=3D'fo=
nt-family:
Tahoma'> x) <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =
</span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>X<sup>1</sup><sub>n</sub>(a)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D d<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Assuming <span
style=3D'mso-spacerun:yes'>&nbsp;</span>d a constant.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>We have <o:p></o:p></s=
pan></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>X<sub>n</sub><sup>1</s=
up> =3D
A<sub>n</sub></span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> Cos </span><span style=3D'font-size:22.0pt'>(=
&#955;</span><span
style=3D'font-family:Tahoma'> x)<span style=3D'mso-tab-count:2'>&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>=
&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;..<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>1<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>When x=3D 0, equation 1
becomes<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>X<sub>n</sub>&#8217;(0=
) =3D A<sub>n
</sub></span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'>=3D d <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>A<sub>n<span
style=3D'mso-spacerun:yes'>&nbsp; </span></sub><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;</span>=3Dd/(</span><span style=3D'f=
ont-size:
16.0pt'>n&#960;</span><span style=3D'font-family:Tahoma'>)<o:p></o:p></span=
></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:4'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></spa=
n></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>X<sub>n</sub>=3D d/(</span><span
style=3D'font-size:22.0pt'> </span><span style=3D'font-size:16.0pt'>n&#960;=
</span><span
style=3D'font-family:Tahoma'>) Sin(</span><span style=3D'font-size:16.0pt'>=
n&#960;</span><span
style=3D'font-family:Tahoma'>x) <span style=3D'mso-tab-count:1'>&nbsp; </sp=
an><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
</span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>n =3D 1,2,3,4,5<span style=3D'mso-t=
ab-count:
1'> </span></span><span style=3D'font-size:22.0pt'><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></=
span></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>Graph<o:p></o:p></spa=
n></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>Example of solution i=
s given
in Fig 1.1.2<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>It passes through 0, =
and 1 as
expected.<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1068" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:9;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:9'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1068" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1029" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image008.png" o:title=3D"relati=
vistic no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image009.gif" v:shapes=3D"_x0000_i1029"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Fig 1.1.2<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.1.2<o:p></o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:18.0pt'>Spread of Particles<o:p></o:p></span></b></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>From diagram fig 1.1.=
3 it can
be seen that the particle has two shapes for values of d=3D2 and d=3D3 at t=
he same
energy, frequency and mass. This means that the particle spreads as Einstein
actually predicted without proof. This means that photons, electrons do not
exist as thin waves but wave spread.<span style=3D'mso-spacerun:yes'>&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1067" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:8;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:8'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1067" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1030" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image010.png" o:title=3D"relati=
vistic spread no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image011.gif" v:shapes=3D"_x0000_i1030"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.1.3<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Case 2 <o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Table 2<o:p></o:p></span></p>

<table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0
 style=3D'border-collapse:collapse;border:none;mso-border-alt:solid windowt=
ext .5pt;
 mso-yfti-tbllook:480;mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-border-inside=
h:
 .5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Function <o:p></o:p=
></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Dirichlet&#8217;s B=
oundary
  Conditions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Neumann Initial Con=
ditions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Hermitian Condition=
<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:1'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(0)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:2'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(a)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:3'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(0)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(a)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>d<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>d<o:p></o:p></span>=
</p>
  </td>
 </tr>
</table>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>From table 2 we see t=
hat the
Hermitian conditions for the existence of solution with real values of ener=
gy
as<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(0)=3DX(a)=3D0 <o:p>=
</o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(a)=3Dd<o:p><=
/o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>This means that we se=
ek
solutions of the equation in the form<span style=3D'mso-spacerun:yes'>&nbsp;
</span>as seen in Fig1.2.1; that is the Eigen function passes through x=3D0=
 and
x=3Da where a=3D4 in this case<o:p></o:p></span></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1071" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:10;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:10'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1071" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1031" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image006.png" o:title=3D"graph =
fig 1 or 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image007.gif" v:shapes=3D"_x0000_i1031"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;</span>Fig
1.2.1<o:p></o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:20.0pt'><o:p>&nbsp;</o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:20.0pt'><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nb=
sp;
</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>Fig
1.2.1<o:p></o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:20.0pt'><o:p>&nbsp;</o:p></span></b></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:20.0pt'>Solution of equation<o:p></o:p></span></b></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>For simplicity of wor=
king a
is chosen as 1<o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><u><span
style=3D'font-size:14.0pt;font-family:Tahoma'><o:p><span style=3D'text-deco=
ration:
 none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Y=3DX(x)T(t)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>In
the case of the time-independent equation this reduces to <o:p></o:p></span=
></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Y=3DX<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Hence
we replace X by Y<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>This gives Hermitian conditions </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </=
span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>X(0)=3DX(1)=
=3D0</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>X&#8217;(1)=3Dd<span style=3D'font-family:Tahoma'><o:p></o:p></span>=
</p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>If X=
(0)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0 and X(1)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>We o=
btain
from solution of equations <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>X
=3D Asin</span><span style=3D'font-size:22.0pt'>(&#955;</span><span
style=3D'font-family:Tahoma'> X) + BCos(</span><span style=3D'font-size:22.=
0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> x)<span style=3D'mso-spacerun:yes'>&nbsp;
</span>--------------- <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>(1)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>X<su=
p>1</sup>=3D
B</span><span style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-f=
amily:
Tahoma'> sin(</span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> x)<span style=3D'mso-spacerun:yes'>&nbsp;&nbs=
p;&nbsp;
</span>+ A </span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> Cos(</span><span style=3D'font-size:22.0pt'>&=
#955;</span><span
style=3D'font-family:Tahoma'> x) ------ (2)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Putt=
ing in
the value x=3D0 in line 1 gives B=3D0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Also=
 by
substituting x=3D1 in X(1)<span style=3D'mso-spacerun:yes'>&nbsp; </span>=
=3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Give=
s <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>X(1)
=3D<span style=3D'mso-spacerun:yes'>&nbsp; </span>Asin</span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
 =3D0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Hence<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:2=
2.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> =3D</span><span style=3D'font-size:16.0pt'> n=
&#960;</span><span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p=
></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>X<sub>n</sub> =
=3D A<sub>n</sub>
*sin</span><span style=3D'font-size:22.0pt'>(&#955;</span><span style=3D'fo=
nt-family:
Tahoma'> x) <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =
</span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>X<sup>1</sup><sub>n</sub>(a)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D d<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>Assuming<span
style=3D'mso-spacerun:yes'>&nbsp; </span>d a constant.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>We have <o:p></o=
:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>X<sub>n</sub><sup>1</s=
up> =3D
A<sub>n</sub></span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> Cos </span><span style=3D'font-size:22.0pt'>(=
&#955;</span><span
style=3D'font-family:Tahoma'> x)<span style=3D'mso-tab-count:2'>&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>=
&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;..<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>1<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>When x=3D 0, equation 1
becomes<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>X<sub>n</sub>&#8217;(1=
) =3D A<sub>n
</sub></span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'>=3D d <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>A<sub>n<span
style=3D'mso-spacerun:yes'>&nbsp; </span></sub><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;</span>=3Dd/(</span><span style=3D'f=
ont-size:
16.0pt'>n&#960;</span><span style=3D'font-family:Tahoma'>)<o:p></o:p></span=
></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:4'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></spa=
n></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>X<sub>n</sub>=3D d/(</=
span><span
style=3D'font-size:22.0pt'> </span><span style=3D'font-size:16.0pt'>n&#960;=
</span><span
style=3D'font-family:Tahoma'>) Sin(</span><span style=3D'font-size:16.0pt'>=
n&#960;</span><span
style=3D'font-family:Tahoma'>x) <span style=3D'mso-tab-count:1'>&nbsp; </sp=
an><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;</span>n
=3D 1,2,3,4,5<span style=3D'mso-tab-count:1'> </span></span><span style=3D'=
font-size:
22.0pt'><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></=
span></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>Graph<o:p></o:p></spa=
n></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>Example of solution i=
s given
in Fig 1.2.2<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>It passes through 0, =
and 1.<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1073" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:12;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:12'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1073" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1032" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image008.png" o:title=3D"relati=
vistic no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image012.gif" v:shapes=3D"_x0000_i1032"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;</span>Fig
1.2.2<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:18.0pt'>Spread of Particles<o:p></o:p></span></b></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>From diagram fig 1.2.=
3 once
more it can be seen that the particle has two shapes for a given values of =
d energy,
frequency and mass. This means that photons, electrons do not exist as thin
waves but wave spread.<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1072" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:11;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:11'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1072" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1033" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image010.png" o:title=3D"relati=
vistic spread no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image013.gif" v:shapes=3D"_x0000_i1033"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Fig 1.2.3<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Case 3<o:p></o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Table 3<o:p></o:p></s=
pan></p>

<table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0
 style=3D'border-collapse:collapse;border:none;mso-border-alt:solid windowt=
ext .5pt;
 mso-yfti-tbllook:480;mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-border-inside=
h:
 .5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Function <o:p></o:p=
></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Dirichlet&#8217;s B=
oundary
  Conditions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Neumann Initial ond=
itions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Hermitian Condition=
<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:1'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(0)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:2'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(a)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:3'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(0)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>d<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>d<o:p></o:p></span>=
</p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(a)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
 </tr>
</table>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>From table 3 we see t=
hat the
Hermitian conditions for the existence of solution with real values of ener=
gy
as<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(0)=3D0 X&#8217;(0)=
=3Dd <o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(a)=3D0<o:p></o:p></=
span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>This means that we se=
ek
solutions of the equation in the form<span style=3D'mso-spacerun:yes'>&nbsp;
</span>as seen in Fig1.2.1; that is the Eigen function passes through x=3D0=
 and
the derivative at x=3D0 is constant d where a=3D4 in this case, and <o:p></=
o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(a)=3D0<o:p><=
/o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1074" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:13;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:13'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1074" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1034" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image014.png" o:title=3D"graph =
 fig 3"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image015.gif" v:shapes=3D"_x0000_i1034"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Fig 1.3.1<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><u><span
style=3D'font-family:Tahoma'>Case III</span></u><span style=3D'font-family:=
Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:20.0pt'>Solution of equation<o:p></o:p></span></b></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>We seek solutions und=
er the
conditions<o:p></o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>X(0)=3DX&#8217;(1)=3D0</p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'>X&#8217;(0)=3Dd<span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>If
X(0)<span style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0 and X1(1)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Solution
of equation gives: <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>X
=3D A Cos</span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> x + B sin </span><span style=3D'font-size:22.=
0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> X<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>X<sup>1</sup>=3D
A</span><span style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-f=
amily:
Tahoma'> sin </span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> x + B </span><span style=3D'font-size:22.0pt'=
>&#955;</span><span
style=3D'font-family:Tahoma'> Cos</span><span style=3D'font-size:22.0pt'>&#=
955;</span><span
style=3D'font-family:Tahoma'> x<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Substituting
X (0) =3D 0<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>X&#82=
17;(1)
=3D<span style=3D'mso-spacerun:yes'>&nbsp; </span>0 we obtain<o:p></o:p></s=
pan></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>0
=3D A Cos0 + B Sin 0<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;
</span>--------------------------- (1)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>0
=3D- </span><span style=3D'font-size:22.0pt'>&#955;</span><span style=3D'fo=
nt-family:
Tahoma'> A sin </span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> + B </span><span style=3D'font-size:22.0pt'>&=
#955;</span><span
style=3D'font-family:Tahoma'> Cos </span><span style=3D'font-size:22.0pt'>&=
#955;<span
style=3D'mso-spacerun:yes'>&nbsp; </span>---- (2)</span><span style=3D'font=
-family:
Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>From
1 it is seen that A=3D0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>From
line 2 we get<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>B</span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
Cos</span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
 =3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:2=
2.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> =3D (<u>2n+1)/2</u></span><span style=3D'font=
-size:
22.0pt'> &#960;</span><span style=3D'font-family:Tahoma'><o:p></o:p></span>=
</p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>X<sub>n</sub>
=3D B<sub>n</sub> Sin (<u>2n+1</u>) </span><span style=3D'font-size:16.0pt'=
>&#960;</span><span
style=3D'font-size:16.0pt;font-family:Tahoma'> </span><span style=3D'font-f=
amily:
Tahoma'>x <span style=3D'mso-tab-count:4'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>4<o:p></o:p=
></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>X<sub>n</sub><sup>1</sup>(0)
=3D d<span style=3D'mso-spacerun:yes'>&nbsp; </span>a<span
style=3D'mso-spacerun:yes'>&nbsp; </span>constant<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Differentiating<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>X<sub>n</sub><sup>1</sup>
=3D B<sub>n</sub><u>2n+1/2</u></span><span style=3D'font-size:22.0pt'> </sp=
an><span
style=3D'font-size:16.0pt'>&#960;</span><span style=3D'font-size:16.0pt;fon=
t-family:
Tahoma'> </span><span style=3D'font-family:Tahoma'>Cos (<u>2n+1</u>)/2 </sp=
an><span
style=3D'font-size:16.0pt'>&#960;</span><span style=3D'font-family:Tahoma'>=
 x<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Substituting
Xn1(0) =3D d in line 4<span style=3D'mso-spacerun:yes'>&nbsp; </span>yields=
<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p=
></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>B<sub>n</sub>
=3D<span style=3D'mso-spacerun:yes'>&nbsp; </span><u><span
style=3D'mso-spacerun:yes'>&nbsp;</span>2d/</u>(2n+1)</span><span
style=3D'font-size:16.0pt'>&#960;</span><u><span style=3D'font-family:Tahom=
a'><o:p></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>X<sub>n</sub>
=3D<u> 2d/(2n+1</u>)</span><span style=3D'font-size:22.0pt'> </span><span
style=3D'font-size:16.0pt'>&#960;</span><span style=3D'font-size:22.0pt'>)<=
/span><u><span
style=3D'font-family:Tahoma'> </span></u><span style=3D'font-family:Tahoma'=
>sin (<u>2n+1</u>)</span><span
style=3D'font-size:22.0pt'> </span><span style=3D'font-size:16.0pt'>&#960;<=
/span><span
style=3D'font-family:Tahoma'>x /2<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>n =3D 0,1,=
2 ------<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>Graph<o:p></o:p></spa=
n></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>See the example of so=
lution
is given in Fig 1.3.2<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>It passes through 0, =
and 1.as
expected.<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1075" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:14;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:14'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1075" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1035" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image016.png" o:title=3D"relati=
vistic no 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image017.gif" v:shapes=3D"_x0000_i1035"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span>Fig 1.3.2<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span><b
style=3D'mso-bidi-font-weight:normal'>Spread of Particles</b><o:p></o:p></s=
pan></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>From diagram fig 1.3.=
3 again
it can be seen that the particle has two shapes for values of d=3D2 and d=
=3D3 at
the same energy, frequency and mass. This means that the particle spreads as
Einstein actually predicted without proof. This means that photons, electro=
ns
do not exist as thin waves but wave spread.<span
style=3D'mso-spacerun:yes'>&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1076" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:15;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:15'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1076" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1036" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image018.png" o:title=3D"relati=
vistic spread no 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image019.gif" v:shapes=3D"_x0000_i1036"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p=
></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Fig 1.3.<o:p></o:p></=
span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Case 4<o:p></o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>We also note that cas=
e 3 and
case gve the same wave function<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Table 4<o:p></o:p></span></p>

<table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0
 style=3D'border-collapse:collapse;border:none;mso-border-alt:solid windowt=
ext .5pt;
 mso-yfti-tbllook:480;mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-border-inside=
h:
 .5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Function <o:p></o:p=
></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Dirichlet&#8217;s B=
oundary
  Conditions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Neumann Initial Con=
ditions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Hermitian Condition=
<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:1'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(0)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:2'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(a)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:3'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(0)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>0<o:p></o:p></span>=
</p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(a)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>d<o:p></o:p></span>=
</p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>d<o:p></o:p></span>=
</p>
  </td>
 </tr>
</table>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>From table 3 we see t=
hat the
Hermitian conditions for the existence of solution with real values of ener=
gy
as<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(0)=3D0 X&#8217;(0)=
=3Dd <o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(a)=3D0<o:p></o:p></=
span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>This means that we se=
ek
solutions of the equation in the form<span style=3D'mso-spacerun:yes'>&nbsp;
</span>as seen in Fig1.2.1; that is the Eigen function passes through x=3D0=
 and
the derivative at x=3D0 is constant d where a=3D4 in this case, and <o:p></=
o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(a)=3Dd<o:p><=
/o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1043" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:2;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:2'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1043" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1037" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image020.png" o:title=3D"graph =
fig 4"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image021.gif" v:shapes=3D"_x0000_i1037"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span>Fig 1.4.1<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;</span><o:p></o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:20.0pt'>Solution of equation<o:p></o:p></span></b></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Here we seek solution=
 under
the condition<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal>X(1)=3DX&#8217;01)=3D0</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'>X&#8217;(1)=3Dconst<span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>In
case IV, we set X(x) =3D 0 at x =3D 0 and X(a) =3D 0 at a =3D 1 in equation=
s below <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>X =3D B Cos</span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> x + A sin </span><span style=3D'font-size:22.=
0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> X<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp; </span>X<sup>1</sup>=3D<span
style=3D'mso-spacerun:yes'>&nbsp; </span>&#955; Bsin </span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
 x + A</span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
Cos</span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
 x<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>gives<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span>X<sub>1</sub>(0) =3D -</span><span style=3D'font-size:22.0pt'>&#955;=
</span><span
style=3D'font-family:Tahoma'> A sin 0 + B </span><span style=3D'font-size:2=
2.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> Cos 0 =3D 0<span style=3D'mso-tab-count:2'>&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>B=3D0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p=
></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span>Hence X =3D A sin </span><span style=3D'font-size:22.0pt'>&#955;</sp=
an><span
style=3D'font-family:Tahoma'> x<span style=3D'mso-tab-count:9'>&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Putting X(x) =3D 0 at x =3D b =3D 1 we obtain<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>X =3D A sin </span><span style=3D'font-size:22.0pt'>&#955;</span><sp=
an
style=3D'font-family:Tahoma'> =3D 0 <span style=3D'mso-tab-count:5'>&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp; </span>36<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:2=
2.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>&#95=
5;</span><span
style=3D'font-family:Tahoma'> =3D <u>n</u></span><span style=3D'font-size:1=
6.0pt'>&#960;</span><span
style=3D'font-family:Tahoma'><span style=3D'mso-tab-count:2'>&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>n
=3D 0, 1, 2, ..<span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp; </span>37<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span>Hence <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span>X =3D A sin(<u>n</u></span><span style=3D'font-size:16.0pt'>&#960;</=
span><span
style=3D'font-family:Tahoma'>x)<span style=3D'mso-tab-count:3'>&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>n
=3D 0,<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>1, 2, 3, .</span=
><span
style=3D'font-size:22.0pt'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span>From <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span>X =3D A sin </span><span style=3D'font-size:22.0pt'>&#955;</span><sp=
an
style=3D'font-family:Tahoma'> x<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;
</span>X<sup>1</sup>(1) =3Dd<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;
</span>constant<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;
</span>We obtain by differentially and setting <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>a=3D1 <o:p=
></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>X<sup>1</sup> =
=3D </span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
 B<sub>2</sub>
cos</span><span style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font=
-family:
Tahoma'> x <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp; </span></span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
 =3D <u>n</u></span><span
style=3D'font-size:16.0pt'>&#960;</span><span style=3D'font-family:Tahoma'>=
<span
style=3D'mso-tab-count:6'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>
<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
</span><span
style=3D'mso-tab-count:1'> </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>A<sub>n</sub> =
=3Dd/<u> n</u></span><span
style=3D'font-size:16.0pt'>&#960;</span><span style=3D'font-family:Tahoma'>=
<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>X =3D A sin <u>n</u></span><span style=3D'font-size:16.0pt'>&#960;</=
span><span
style=3D'font-family:Tahoma'> x<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>Graph<o:p></o:p></spa=
n></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>Once more example of =
solution
is given in Fig 1.4.2<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>It passes through 0, =
and 1 as
expected..<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1110" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:29;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:29'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1110" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1038" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image008.png" o:title=3D"relati=
vistic no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image009.gif" v:shapes=3D"_x0000_i1038"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.4.2<o:p></o:p></span></p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:18.0pt'>Spread of Particles<o:p></o:p></span></b></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>From diagram fig 1..4=
.3 to
say it one more time it can be seen that the particle has two shapes for va=
lues
of d=3D2 and d=3D3 at the same energy, frequency and mass. This means that =
the
particle spreads as Einstein actually predicted without proof. This means t=
hat
photons, electrons do not exist as thin waves but wave spread. <o:p></o:p><=
/span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1111" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:30;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:30'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1111" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1039" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image010.png" o:title=3D"relati=
vistic spread no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image022.gif" v:shapes=3D"_x0000_i1039"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Fig 1.4.3<o:p></o:p><=
/span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Summary of work<o:p><=
/o:p></span></p>

<p class=3Dparagraph><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>(=
1) For a=3DL
the energy is given as &nbsp;<o:p></o:p></span></p>

<p class=3Dparagraph><b style=3D'mso-bidi-font-weight:normal'><sub><span
lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-language:EN-US;
mso-no-proof:yes'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_5" o:spid=3D"_x0000_=
i1040"
 type=3D"#_x0000_t75" style=3D'width:141pt;height:33pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image023.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D188 height=3D44
src=3D"relativistic_files/image023.gif" v:shapes=3D"Picture_x0020_5"><![end=
if]></span></sub></b><span
lang=3DEN-US style=3D'mso-ansi-language:EN-US'><o:p></o:p></span></p>

<p class=3Dparagraph><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>It can be =
seen
that while old solution<span style=3D'mso-ansi-language:EN-US'> <span lang=
=3DEN-US><o:p></o:p></span></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><i
style=3D'mso-bidi-font-style:normal'><span style=3D'font-family:Tahoma'><o:=
p>&nbsp;</o:p></span></i></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><i
style=3D'mso-bidi-font-style:normal'><span style=3D'font-family:Tahoma'>TAB=
LE OF
SOLUTIONS<o:p></o:p></span></i></p>

<table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0
 style=3D'border-collapse:collapse;border:none;mso-border-alt:solid windowt=
ext .5pt;
 mso-yfti-tbllook:480;mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-border-inside=
h:
 .5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;mso-border-alt:so=
lid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>CASE<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-left:none;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>HERMITIAN
  CONDITIONS<o:p></o:p></span></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border:solid windowtex=
t 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>NEW
  SOLUTION<o:p></o:p></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border:solid windowte=
xt 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>ENERGY=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<span
  style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
  </span><o:p></o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:1'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-top:none;
  mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext =
.5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>1<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border-top:none;border-left:none;border-bottom:=
solid windowtext 1.0pt;
  border-right:solid windowtext 1.0pt;mso-border-top-alt:solid windowtext .=
5pt;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal>X(0)=3DX(1)=3D0<span lang=3DEN-US style=3D'mso-ansi-=
language:EN-US;
  mso-fareast-language:EN-US'><o:p></o:p></span></p>
  <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'>X&#8217;(0)=3Dconstant<s=
pan
  style=3D'font-family:Tahoma'><o:p></o:p></span></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border-top:none;border=
-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p><span lang=3DEN-US>(d/n&#960;)*sin((n&#960;x)</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-fon=
t-weight:
  normal'>En=3D</b><span style=3D'font-size:22.0pt'> (n<sup>2</sup>&#960;<s=
up>2</sup>
  h<sup>2</sup> )/(2m)</span><span style=3D'font-family:Tahoma'><o:p></o:p>=
</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:2'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-top:none;
  mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext =
.5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>2<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border-top:none;border-left:none;border-bottom:=
solid windowtext 1.0pt;
  border-right:solid windowtext 1.0pt;mso-border-top-alt:solid windowtext .=
5pt;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal>X&#8217;(0)=3DX(1)=3D0<span lang=3DEN-US style=3D'ms=
o-ansi-language:
  EN-US;mso-fareast-language:EN-US'><o:p></o:p></span></p>
  <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
  <p class=3DMsoNormal>X&#8217;(1)=3Dcons<span style=3D'font-size:18.0pt'><=
o:p></o:p></span></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border-top:none;border=
-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p><span lang=3DEN-US>(d/n&#960;)*sin((n&#960;x)</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-fon=
t-weight:
  normal'>En=3D</b><span style=3D'font-size:22.0pt'> (k<sup>2</sup>&#960;<s=
up>2</sup>
  h<sup>2</sup> )/(2m)</span><span style=3D'font-family:Tahoma'><o:p></o:p>=
</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:3'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-top:none;
  mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext =
.5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>3<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border-top:none;border-left:none;border-bottom:=
solid windowtext 1.0pt;
  border-right:solid windowtext 1.0pt;mso-border-top-alt:solid windowtext .=
5pt;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal>X(0)=3DX&#8217;(1)=3D0<span lang=3DEN-US style=3D'ms=
o-ansi-language:
  EN-US;mso-fareast-language:EN-US'><o:p></o:p></span></p>
  <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'>X&#8217;(0)=3Dconstant<s=
pan
  style=3D'font-family:Tahoma'><o:p></o:p></span></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border-top:none;border=
-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p><span lang=3DEN-US>(d/k&#960;)*sin((k&#960;x)</span></p>
  <p><span lang=3DEN-US style=3D'font-family:Tahoma'>k=3D(2n+1)/2<o:p></o:p=
></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-fon=
t-weight:
  normal'>En=3D</b><span style=3D'font-size:22.0pt'> (n<sup>2</sup>&#960;<s=
up>2</sup>
  h<sup>2</sup> )/(2m)</span><span style=3D'font-family:Tahoma'><o:p></o:p>=
</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>k=3D(2n+1)/2<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-top:none;
  mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext =
.5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'>F<span style=3D'font-fam=
ily:Tahoma'>4<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border-top:none;border-left:none;border-bottom:=
solid windowtext 1.0pt;
  border-right:solid windowtext 1.0pt;mso-border-top-alt:solid windowtext .=
5pt;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal>X&#8217;(0)=3DX(1)=3D0<span lang=3DEN-US style=3D'ms=
o-ansi-language:
  EN-US;mso-fareast-language:EN-US'><o:p></o:p></span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'>X&#8217;(1)=3Dconstant<s=
pan
  style=3D'font-family:Tahoma'><o:p></o:p></span></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border-top:none;border=
-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p><span lang=3DEN-US>(d/n&#960;)*sin((n&#960;x)</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-fon=
t-weight:
  normal'>En=3D</b><span style=3D'font-size:22.0pt'>((k<sup>2</sup>&#960;<s=
up>2</sup>
  h<sup>2</sup>)/(2m)<o:p></o:p></span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
 </tr>
</table>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span>Table 5<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;</span>of Schrodinger&#8217;s equation giv=
es one
wave function the new one gives two distinct solutions.<span
style=3D'mso-spacerun:yes'>&nbsp; </span>Cases 1, 2, and 4 give the same
eigenfuncion <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;
</span>Y =3D A sin <u>n</u></span><span style=3D'font-size:16.0pt'>&#960;</=
span><span
style=3D'font-family:Tahoma'>x<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>while
case 3 gives different eigenfunction<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>.
X<sub>n</sub> =3D<u> 2d/(2n+1</u>)</span><span style=3D'font-size:22.0pt'> =
</span><span
style=3D'font-size:16.0pt'>&#960;</span><span style=3D'font-size:22.0pt'>)<=
/span><u><span
style=3D'font-family:Tahoma'> </span></u><span style=3D'font-family:Tahoma'=
>sin (<u>2n+1</u>)</span><span
style=3D'font-size:22.0pt'> </span><span style=3D'font-size:16.0pt'>&#960;<=
/span><span
style=3D'font-family:Tahoma'>x /2<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>n =3D 0,1,=
2 ------<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-font-=
weight:
normal'><span style=3D'font-size:16.0pt;font-family:Tahoma'>Graph<o:p></o:p=
></span></b></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>See
graph fig 1.5.1 below which shows two waves..</span><span style=3D'font-siz=
e:
18.0pt'><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1113" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:31;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:31'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1113" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1041" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image010.png" o:title=3D"relati=
vistic spread no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image024.gif" v:shapes=3D"_x0000_i1041"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1114" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:32;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:32'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1114" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1042" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image018.png" o:title=3D"relati=
vistic spread no 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image025.gif" v:shapes=3D"_x0000_i1042"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span>Fig 1.3<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:14.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>Fig
1.5.1<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>Electron co=
nsists
of two waves.<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:1=
8.0pt'>More
will be said on this in the next paper.<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Fig 1.4.3<o:p></o:p><=
/span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:1=
8.0pt'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:1=
8.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;</span></span><span style=3D'font-family:T=
ahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1105" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:-36pt;margin-top:13.05pt;width:242.=
4pt;
 height:235.2pt;z-index:26;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout;positi=
on:
absolute;z-index:26;margin-left:-49px;margin-top:16px;width:329px;height:32=
0px'>

<table cellpadding=3D0 cellspacing=3D0>
 <tr>
  <td width=3D329 height=3D320 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:26'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1105" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1043" ty=
pe=3D"#_x0000_t75"
     style=3D'width:227.25pt;height:227.25pt'>
     <v:imagedata src=3D"relativistic_files/image026.png" o:title=3D"graph =
n equals 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D303 height=3D303
    src=3D"relativistic_files/image027.gif" v:shapes=3D"_x0000_i1043"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><!--[if gte vml 1]><v:shape id=3D"_x0000_s1096" type=3D"#_=
x0000_t202"
 style=3D'position:absolute;margin-left:225pt;margin-top:8.35pt;width:233.4=
pt;
 height:226.2pt;z-index:25;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout;positi=
on:
absolute;z-index:25;margin-left:299px;margin-top:10px;width:317px;height:30=
8px'>

<table cellpadding=3D0 cellspacing=3D0>
 <tr>
  <td width=3D317 height=3D308 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:25'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1096" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1044" ty=
pe=3D"#_x0000_t75"
     style=3D'width:218.25pt;height:218.25pt'>
     <v:imagedata src=3D"relativistic_files/image028.png" o:title=3D"graph =
n equals 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D291 height=3D291
    src=3D"relativistic_files/image029.gif" v:shapes=3D"_x0000_i1044"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;
</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:1=
8.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Fig 1.4.4<=
span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;
</span>fig 1.4.5<span style=3D'mso-spacerun:yes'>&nbsp; </span></span><span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Thank you. <o:p></o:p=
></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p><span class=3Dgoogqs-tidbit1><span lang=3DEN-US>In order to extend
Schr&ouml;dinger's formalism to include relativity, the physical picture mu=
st
be altered. The<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>Klein-G=
orden
equation </span></span><span lang=3DEN-US>uses the relativistic mass-energy
relation:</span></p>

<p class=3DMsoNormal style=3D'margin-left:36.0pt'><span style=3D'mso-no-pro=
of:yes'><!--[if gte vml 1]><v:shape
 id=3D"_x0000_i1045" type=3D"#_x0000_t75" alt=3D"E^2 =3D p^2c^2 + m^2c^4\,"=
 style=3D'width:111pt;
 height:16.5pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image004.gif" o:title=3D","/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D148 height=3D22
src=3D"relativistic_files/image004.gif" alt=3D"E^2 =3D p^2c^2 + m^2c^4\," v=
:shapes=3D"_x0000_i1045"><![endif]></span></p>

<p><span lang=3DEN-US>to produce the differential equation<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span></span></p>

<p class=3DMsoNormal style=3D'margin-left:36.0pt'><span style=3D'mso-no-pro=
of:yes'><!--[if gte vml 1]><v:shape
 id=3D"_x0000_i1046" type=3D"#_x0000_t75" alt=3D"- \frac{1}{c^2}{\partial^2=
 \over \partial t^2}\psi =3D  - \nabla^2 \psi + \frac{m^2c^2}{\hbar^2} \psi=
\,"
 style=3D'width:180pt;height:33pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image005.gif" o:title=3D","/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D240 height=3D44
src=3D"relativistic_files/image005.gif"
alt=3D"- \frac{1}{c^2}{\partial^2 \over \partial t^2}\psi =3D  - \nabla^2 \=
psi + \frac{m^2c^2}{\hbar^2} \psi\,"
v:shapes=3D"_x0000_i1046"><![endif]></span></p>

<p><span lang=3DEN-US>which is relativistically invariant.</span></p>

<p><span lang=3DEN-US>As before we replace the Greek letter by Y(x,t).</spa=
n></p>

<p><span lang=3DEN-US>Also introducing that x, and t are independent and al=
so
Y(x,t)<span style=3D'mso-spacerun:yes'>&nbsp; </span>is seperable</span></p>

<p><span lang=3DEN-US>Y(x,t)=3DX(x)T(t)</span></p>

<p><span lang=3DEN-US>Where X is a space variable and T is time<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span></span></p>

<p><span lang=3DEN-US>Also denoting </span></p>

<p><span lang=3DEN-US style=3D'font-size:16.0pt'>X<sub>xx</sub></span><span
lang=3DEN-US> as derivatives with respect to x</span></p>

<p><span lang=3DEN-US style=3D'font-size:16.0pt'>T<sub>tt</sub></span><span
lang=3DEN-US><span style=3D'mso-spacerun:yes'>&nbsp; </span>as derivatives =
with
respect to t (time)</span></p>

<p class=3DMsoNormal>The equation above reduces to two equations given as</=
p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'mso-spacerun:yes'>&nbsp;</span><span
style=3D'font-size:16.0pt'>X<sub>xx<span style=3D'mso-spacerun:yes'>&nbsp; =
</span>+
</sub><span style=3D'mso-spacerun:yes'>&nbsp;</span>EX=3D0</span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>T<sub>tt + </sub>k<su=
p>2</sup>
T<sub>tt<span style=3D'mso-spacerun:yes'>&nbsp; </span></sub>=3D0</span><su=
b><span
style=3D'font-size:20.0pt'><o:p></o:p></span></sub></p>

<p class=3DMsoNormal>Assuming V is small or zero and <span style=3D'font-si=
ze:16.0pt'>k<sup>2</sup></span>=3D<span
style=3D'font-size:16.0pt'> C<sup>2</sup></span> (E+ <span style=3D'font-si=
ze:16.0pt'>m<sub>e</sub><sup>2</sup>
C<sup>2/</sup> h<sup>2 </sup></span>)</p>

<p class=3DMsoNormal>That is </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>k=
=3D+or
&#8211; square root(<span style=3D'font-size:16.0pt'>C<sup>2</sup></span> (=
E+ <span
style=3D'font-size:16.0pt'>m<sub>e</sub><sup>2</sup> C<sup>2/</sup> h<sup>2=
 </sup></span>)</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>We also assume that the final solution will pass throu=
gh
points </p>

<p class=3DMsoNormal>Which are multiples of wavelengths of the waves. At any
point the time is distance divided by the speed of light.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>We get solutions of the form</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'font-size:16.0pt'>T(t)=3DA. sin(kt)<o:p=
></o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D------------------------------------------------------------------------=
------<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Gallery<o:p></o:p></s=
pan></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0
 style=3D'border-collapse:collapse;border:none;mso-border-alt:solid windowt=
ext .5pt;
 mso-yfti-tbllook:480;mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-border-inside=
h:
 .5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Function <o:p></o:p=
></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Dirichlet&#8217;s B=
oundary
  conditions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Neumann initial con=
ditions<o:p></o:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border:solid windowt=
ext 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>Hermitian Condition=
<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:1'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(0)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:2'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X(a)<o:p></o:p></sp=
an></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:3'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(0)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border:solid windowte=
xt 1.0pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:s=
olid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'>X&#8217;(a)<o:p></o=
:p></span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.5pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
  <td width=3D142 valign=3Dtop style=3D'width:106.55pt;border-top:none;bord=
er-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></=
span></p>
  </td>
 </tr>
</table>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1044" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:3;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:3'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1044" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1047" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image030.png" o:title=3D"graph =
fig 5 non relativistic solution"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image031.gif" v:shapes=3D"_x0000_i1047"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;
</span>Fig 1.4<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;</span><o:p></o:p></span=
></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1045" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:4;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:4'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1045" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1048" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image032.png" o:title=3D"graph =
second wave non relativistic"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image033.gif" v:shapes=3D"_x0000_i1048"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.5<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1046" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:5;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:5'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1046" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1049" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image034.png" o:title=3D"graph =
spread example"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image035.gif" v:shapes=3D"_x0000_i1049"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.6<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1047" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:6;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:6'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1047" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1050" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image036.png" o:title=3D"graph =
spread of electron second wave"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image037.gif" v:shapes=3D"_x0000_i1050"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;</span>Fig =
1.7<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1048" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:.9pt;width:251.4pt;
 height:244.2pt;z-index:7;mso-wrap-style:none'>
 <v:textbox style=3D'mso-fit-shape-to-text:t'/>
</v:shape><![endif]--><![if !vml]><span style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D0></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D332 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:7'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1048" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_i1051" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"relativistic_files/image038.png" o:title=3D"graph =
electron as two waves"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"relativistic_files/image039.gif" v:shapes=3D"_x0000_i1051"><![en=
dif]></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.8<o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1090" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:108pt;margin-top:6.3pt;width:243pt;
 height:207pt;z-index:21'/><![endif]--><![if !vml]><span style=3D'mso-ignor=
e:
vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D143 height=3D7></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D330 height=3D282 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:21'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1090" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></span><=
/p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p><span class=3Dgoogqs-tidbit1><span lang=3DEN style=3D'mso-ansi-language:=
EN'>In
order to extend Schr&ouml;dinger's formalism to include relativity, the
physical picture must be altered. The<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>Klein-Gorden equation </span=
></span><span
lang=3DEN style=3D'mso-ansi-language:EN'>uses the relativistic mass-energy
relation:<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'margin-left:36.0pt'><span lang=3DEN style=3D'=
mso-ansi-language:
EN'><!--[if gte vml 1]><v:shape id=3D"_x0000_i1052" type=3D"#_x0000_t75" al=
t=3D"E^2 =3D p^2c^2 + m^2c^4\,"
 style=3D'width:111pt;height:16.5pt'>
 <v:imagedata src=3D"relativistic_files/image004.gif" o:href=3D"http://uplo=
ad.wikimedia.org/math/8/2/9/829beaa09287f661d88f6ad23161f777.png"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D148 height=3D22
src=3D"relativistic_files/image004.gif" alt=3D"E^2 =3D p^2c^2 + m^2c^4\," v=
:shapes=3D"_x0000_i1052"><![endif]><o:p></o:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>to produce the differenti=
al equation<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'margin-left:36.0pt'><span lang=3DEN style=3D'=
mso-ansi-language:
EN'><!--[if gte vml 1]><v:shape id=3D"_x0000_i1053" type=3D"#_x0000_t75" al=
t=3D"- \frac{1}{c^2}{\partial^2 \over \partial t^2}\psi =3D  - \nabla^2 \ps=
i + \frac{m^2c^2}{\hbar^2} \psi\,"
 style=3D'width:180pt;height:33pt'>
 <v:imagedata src=3D"relativistic_files/image005.gif" o:href=3D"http://uplo=
ad.wikimedia.org/math/a/c/2/ac2233fce3429f914c3205da3c5b135b.png"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D240 height=3D44
src=3D"relativistic_files/image005.gif"
alt=3D"- \frac{1}{c^2}{\partial^2 \over \partial t^2}\psi =3D  - \nabla^2 \=
psi + \frac{m^2c^2}{\hbar^2} \psi\,"
v:shapes=3D"_x0000_i1053"><![endif]><o:p></o:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>which is relativistically
invariant.<o:p></o:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>As before we replace the =
Greek
letter by Y.<o:p></o:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>Also introducing <o:p></o=
:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>Y=3DXT<o:p></o:p></span><=
/p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>Where X is a space variab=
le and T
is time <o:p></o:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>Also denoting a bit confu=
sing
symbols<o:p></o:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>X&#8217; X&#8217;&#8217; =
as
derivatives with respect to x<o:p></o:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>And T&#8217; T&#8217;&#82=
17; as
derivatives with respect to t (time)<o:p></o:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'><o:p>&nbsp;</o:p></span><=
/p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'><o:p>&nbsp;</o:p></span><=
/p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'><o:p>&nbsp;</o:p></span><=
/p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'><o:p>&nbsp;</o:p></span><=
/p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'>C2<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>C4<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>M2<span
style=3D'mso-spacerun:yes'>&nbsp; </span>P2<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>E2<span
style=3D'mso-spacerun:yes'>&nbsp; </span>h2<o:p></o:p></span></p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'><o:p>&nbsp;</o:p></span><=
/p>

<p><span lang=3DEN style=3D'mso-ansi-language:EN'><o:p>&nbsp;</o:p></span><=
/p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1091" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:90pt;margin-top:-36pt;width:198pt;
 height:153pt;z-index:22'/><![endif]--><![if !vml]><span style=3D'mso-ignor=
e:
vglayout;position:relative;z-index:22'><span style=3D'position:absolute;
left:119px;top:-49px;width:270px;height:210px'>

<table cellpadding=3D0 cellspacing=3D0>
 <tr>
  <td width=3D270 height=3D210 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:22'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
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    <div v:shape=3D"_x0000_s1091" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    </div>
    <![if !mso]></td>
   </tr>
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  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span></span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p><=
/span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1092" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:90pt;margin-top:-36pt;width:198pt;
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e:
vglayout;position:relative;z-index:23'><span style=3D'position:absolute;
left:119px;top:-49px;width:270px;height:210px'>

<table cellpadding=3D0 cellspacing=3D0>
 <tr>
  <td width=3D270 height=3D210 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:23'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1092" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    </div>
    <![if !mso]></td>
   </tr>
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  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span></span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p><=
/span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

<p class=3DMsoNormal><!--[if gte vml 1]><v:shape id=3D"_x0000_s1093" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:90pt;margin-top:-36pt;width:198pt;
 height:153pt;z-index:24'/><![endif]--><![if !vml]><span style=3D'mso-ignor=
e:
vglayout;position:relative;z-index:24'><span style=3D'position:absolute;
left:119px;top:-49px;width:270px;height:210px'>

<table cellpadding=3D0 cellspacing=3D0>
 <tr>
  <td width=3D270 height=3D210 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:24'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1093" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span></span><![endif]><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p><=
/span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

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an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;
</span><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;</span><o:p></o:p=
></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><span
style=3D'font-family:Tahoma'>A</span> <span style=3D'font-family:Tahoma'>S<b
style=3D'mso-bidi-font-weight:normal'>urvey of Boundary Conditions of
Schrodinger&#8217;s Equation.<o:p></o:p></b></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>3:2<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>SUMM=
ARY.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>For any eigenfunction y which is a solution of
Schrodinger&#8217;s equation on an interval (a, b) to have real eigenvector=
, it
has to satisfy one of a minimum set of four conditions which are proved in =
the
following discussion.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>For simplicity, yet without loss of generality=
, the
interval is chosen as (0, 1)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><!--[if gte vml 1]><v:shap=
e id=3D"_x0000_s1078"
 type=3D"#_x0000_t202" style=3D'position:absolute;margin-left:18pt;margin-t=
op:1.85pt;
 width:129.6pt;height:1in;z-index:17'/><v:shape id=3D"_x0000_s1079" type=3D=
"#_x0000_t202"
 style=3D'position:absolute;margin-left:234pt;margin-top:1.85pt;width:117pt;
 height:1in;z-index:18'/><![endif]--><![if !vml]><span style=3D'mso-ignore:=
vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D23 height=3D1></td>
  <td width=3D179></td>
  <td width=3D109></td>
  <td width=3D162></td>
 </tr>
 <tr>
  <td height=3D102></td>
  <td width=3D179 height=3D102 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:17'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1078" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal>Y(0)=3DY(1)=3D0</p>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    <p class=3DMsoNormal>Y&#8217;(0)=3Dconstant</p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
  <td></td>
  <td width=3D162 height=3D102 bgcolor=3Dwhite style=3D'border:.75pt solid =
black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:18'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1079" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal>Y(0)=3DY&#8217;(1)=3D0</p>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    <p class=3DMsoNormal>Y&#8217;(0)=3Dconstant</p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span=
></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p=
></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span>CONDITION I<span style=3D'mso-tab-count:4'>&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>CONDITION
II<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><!--[if gte vml 1]><v:shap=
e id=3D"_x0000_s1080"
 type=3D"#_x0000_t202" style=3D'position:absolute;margin-left:18pt;margin-t=
op:17.2pt;
 width:126pt;height:81pt;z-index:19'>
 <v:textbox style=3D'mso-next-textbox:#_x0000_s1080'/>
</v:shape><v:shape id=3D"_x0000_s1081" type=3D"#_x0000_t202" style=3D'posit=
ion:absolute;
 margin-left:234pt;margin-top:8.45pt;width:117pt;height:81pt;z-index:20'/><=
![endif]--><![if !vml]><span
style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D23 height=3D10></td>
  <td width=3D174></td>
  <td width=3D114></td>
  <td width=3D162></td>
 </tr>
 <tr>
  <td height=3D12></td>
  <td colspan=3D2></td>
  <td rowspan=3D2 width=3D162 height=3D114 bgcolor=3Dwhite style=3D'border:=
.75pt solid black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:20'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1081" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    <p class=3DMsoNormal>Y&#8217;(0)=3DY(1)=3D0</p>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    <p class=3DMsoNormal>Y&#8217;(1)=3Dconstant</p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
 <tr>
  <td height=3D102></td>
  <td rowspan=3D2 width=3D174 height=3D114 bgcolor=3Dwhite style=3D'border:=
.75pt solid black;
  vertical-align:top;background:white'><![endif]><![if !mso]><span
  style=3D'position:absolute;mso-ignore:vglayout;z-index:19'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1080" style=3D'padding:4.35pt 7.95pt 4.35pt 7.9=
5pt'
    class=3Dshape>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    <p class=3DMsoNormal>Y&#8217;(0)=3DY(1)=3D0</p>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    <p class=3DMsoNormal>Y&#8217;(1)=3Dconstant</p>
    <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
 <tr>
  <td height=3D12></td>
 </tr>
</table>

</span><![endif]><span style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span=
></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span>CONDITION III<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span>CONDITION IV<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><b
style=3D'mso-bidi-font-weight:normal'><u><span style=3D'font-family:Tahoma'=
>The
Complete Boundary Conditions<o:p></o:p></span></u></b></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-font-=
weight:
normal'><span style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></b></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>In
the previous discussion it was seen that for the Hamiltonian operator H be
hermit ion it is necessary that the following boundary condition be satisfi=
ed. <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><!--[if gte vml 1]><v:shap=
etype
 id=3D"_x0000_t185" coordsize=3D"21600,21600" o:spt=3D"185" adj=3D"3600" pa=
th=3D"m@0,nfqx0@0l0@2qy@0,21600em@1,nfqx21600@0l21600@2qy@1,21600em@0,nsqx0=
@0l0@2qy@0,21600l@1,21600qx21600@2l21600@0qy@1,xe"
 filled=3D"f">
 <v:formulas>
  <v:f eqn=3D"val #0"/>
  <v:f eqn=3D"sum width 0 #0"/>
  <v:f eqn=3D"sum height 0 #0"/>
  <v:f eqn=3D"prod @0 2929 10000"/>
  <v:f eqn=3D"sum width 0 @3"/>
  <v:f eqn=3D"sum height 0 @3"/>
  <v:f eqn=3D"val width"/>
  <v:f eqn=3D"val height"/>
  <v:f eqn=3D"prod width 1 2"/>
  <v:f eqn=3D"prod height 1 2"/>
 </v:formulas>
 <v:path o:extrusionok=3D"f" gradientshapeok=3D"t" limo=3D"10800,10800"
  o:connecttype=3D"custom" o:connectlocs=3D"@8,0;0,@9;@8,@7;@6,@9" textboxr=
ect=3D"@3,@3,@4,@5"/>
 <v:handles>
  <v:h position=3D"#0,topLeft" switch=3D"" xrange=3D"0,10800"/>
 </v:handles>
</v:shapetype><v:shape id=3D"_x0000_s1077" type=3D"#_x0000_t185" style=3D'p=
osition:absolute;
 margin-left:-18pt;margin-top:3.1pt;width:117pt;height:36pt;z-index:16'/><!=
[endif]--><![if !vml]><span
style=3D'mso-ignore:vglayout;position:absolute;z-index:16;margin-left:-25px;
margin-top:3px;width:158px;height:50px'><img width=3D158 height=3D50
src=3D"relativistic_files/image040.gif" v:shapes=3D"_x0000_s1077"></span><!=
[endif]><span
style=3D'font-family:Tahoma'><span style=3D'mso-tab-count:4'>&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>[Y<sub>1</sub>1Y2<span
style=3D'mso-spacerun:yes'>&nbsp; </span>-<span style=3D'mso-spacerun:yes'>=
&nbsp;
</span>Y<sub>2</sub>1Y1]<span style=3D'mso-spacerun:yes'>&nbsp; </span>=3D =
0<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span><span style=3D'mso-tab=
-count:
7'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>1<o:p></o:=
p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:5'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span=
><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Expanding
the above equation gives<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>1</sub>(a)Y<sub>2</sub><sup>1=
</sup>(a)
&#8211; Y<sub>2</sub>(a)Y<sub>1</sub><sup>1</sup>(a) <span style=3D'mso-tab=
-count:
4'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>2<o:p></o:p></sp=
an></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>+Y<sub>1</sub>(b)Y<sub>2</sub><sup>=
1</sup>(b)
&#8211; Y<sub>2</sub>(a)Y<sub>1</sub><sup>1</sup>(b)<span style=3D'mso-tab-=
count:
1'>&nbsp;&nbsp; </span>=3D 0<span style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>3<o:p></o:p></span>=
</p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>The
above equation is satisfied in a number of ways and four ways are going to =
be
discussed below.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'>Case
0<o:p></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y(a)
=3D d, Y(b) =3D e<span style=3D'mso-tab-count:5'>&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </sp=
an>4<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sup>i</sup>(a)
=3D f(t1),<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>Y<sup>=
i</sup>(b)
=3D g (t)<span style=3D'mso-tab-count:4'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>5<=
o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Where
d, e is any constant and f1 g are either constants of functions <o:p></o:p>=
</span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>of
an arbitrary parameter t.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>To
prove that conditions 4, 5 satisfy,1, 2 and 3 we do a bit of <o:p></o:p></s=
pan></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>substitution
here <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>From
2 and 3 let<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>1</sub>(a)Y<sub>2</sub><sup>1=
</sup>(a)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>- Y<sub>2</sub>(a)Y<sub>1</sub><su=
p>1</sup>(a)
=3D K<span style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>6<o:p></o:p></span=
></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>1</sub>(b)Y<sub>2</sub><sup>1=
</sup>(b)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>- Y<sub>2</sub>(b)Y<sub>1</sub><su=
p>1</sup>(b)
=3D P<span style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>7<o:p></o:p></span=
></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Putting,<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Y (a) =3D d and Y<sub>1</sub>(a)<span style=3D'mso-spacerun:yes'>&nb=
sp;
</span>=3D f(t)<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>k becom=
es<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>K
=3D d f (t) - f (t) d =3D 0<span style=3D'mso-tab-count:4'>&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span=
></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Similarly
putting, Y<sub>1</sub> (b) =3D g (t) and Y(b)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D e p becomes<o:p></o:p></span><=
/p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>P
=3D Y<sub>1</sub>(b)Y<sub>2</sub><sup>1</sup>(b)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>- Y<sub>1</sub><sup>1</sup>(b)Y<su=
b>2</sub>(b)<span
style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>9<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>=3D
e g (t)<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>- g (t) e =3D 0=
 <span
style=3D'mso-tab-count:4'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp; </span>10<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>Here we can see that condition 4 and 5 satisfy
condition 1. the case 0 so far considered is a more general case, but we wi=
ll
still consider what happens to be special cases of case 0.<o:p></o:p></span=
></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><u><span
style=3D'font-family:Tahoma'>Case I<o:p></o:p></span></u></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Suppose
here that, <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y(a)
=3D Y(b) =3D 0 and <span style=3D'mso-tab-count:5'>&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>11<o:p></o:p>=
</span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>n</sub><sup>1</sup>(a)
=3D f<sub>n</sub>(t),<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>n</s=
ub><sup>1</sup>(b)
=3D g<sub>n</sub>(t)<span style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>12<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>Where fn(t)<sub>1</sub><span
style=3D'mso-spacerun:yes'>&nbsp; </span>g<sub>n</sub>(t) are either consta=
nts or
any functions of arbitrary <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>parameter t .<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>If we substitute Y (a) =3D Y (b) =3D 0 into 6 =
and 7 we
obtain<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>K
=3DY<sub>1</sub>(a)Y<sub>2</sub><sup>1</sup>(a)<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>-<span
style=3D'mso-spacerun:yes'>&nbsp; </span>Y<sub>2</sub>(a)Y<sub>1</sub><sup>=
1</sup>(a)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>=3D
0 x Y<sub>2</sub>1(a)<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>-=
<span
style=3D'mso-spacerun:yes'>&nbsp; </span>0x Y<sub>1</sub><sup>1</sup>(a) =
=3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>P=3D
Y<sub>1</sub>(b)Y<sub>2</sub><sup>1</sup>(b)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>- Y<sub>1</sub><sup>1</sup>(b)Y<su=
b>2</sub>(b)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0 x Y<sub>2</sub><sup>1</sup>(=
b)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>- Y<sub>1</sub><sup>1</sup>(b) x 0=
<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D0 <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>So
here again we have established a second condition which satisfies 1<o:p></o=
:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><u><span
style=3D'font-family:Tahoma'>Case II<o:p></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Suppose
this time we try the case when the derivative is equal to zero<span
style=3D'mso-spacerun:yes'>&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>at
x =3D<span style=3D'mso-spacerun:yes'>&nbsp; </span>.<span
style=3D'mso-spacerun:yes'>&nbsp; </span>Let the function vanish at x =3D b=
, i.e.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>1</sub>(a)
=3D 0<span style=3D'mso-spacerun:yes'>&nbsp; </span>Y(b)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0<span style=3D'mso-tab-count:=
5'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp; </span>13<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>and
let<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>1</sub>(b)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D g(t)<span style=3D'mso-tab-cou=
nt:6'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp; </span>14<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>From
substituting in 2 and 3 as in previous cases, we obtain<o:p></o:p></span></=
p>

<p class=3DListParagraphCxSpFirst style=3D'margin-left:0cm;mso-add-space:au=
to;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>K<span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>=3D<span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp; </span>Y<sub>1</sub>(a)Y<sub>2</sub><sup>1</sup>(a)
<span style=3D'mso-spacerun:yes'>&nbsp;</span>&#8211;<span
style=3D'mso-spacerun:yes'>&nbsp; </span>Y<sub>2 </sub>(a)Y<sub>1</sub><sup=
>1</sup>
(a)<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span>=3D<span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp; </span>Y<sub>1</sub>(a)
x0 - Y<sub>2</sub>(a) x0<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span>=3D<span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp; </span>0<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'>and<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'>P<span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp; </span>=3D<span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp; </span>Y<sub>1</sub>(b)Y<sub>2</sub><sup>1</sup>(b)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>&#8211;<span
style=3D'mso-spacerun:yes'>&nbsp; </span>Y<sub>2 </sub>(b)Y<sub>1</sub><sup=
>1</sup>
(b)<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span>=3D<span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp; </span>0xY<sub>1</sub>(a)<sub>
2</sub><sup>1</sup> (b) &#8211; 0xY<sub>1</sub><sup>1</sup> (b)<o:p></o:p><=
/span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span>=3D<span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp; </span>0<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'>Hence
condition 1 is satisfied once more in Case II.<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle align=3Dcenter style=3D'margin-left:0cm;
mso-add-space:auto;text-align:center'><b style=3D'mso-bidi-font-weight:norm=
al'><span
style=3D'font-size:12.0pt;font-family:"Times New Roman"'>Case III<o:p></o:p=
></span></b></p>

<p class=3DListParagraphCxSpMiddle align=3Dcenter style=3D'margin-left:0cm;
mso-add-space:auto;text-align:center'><span style=3D'font-size:12.0pt;font-=
family:
"Times New Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'>Case
III &amp; IV can be proved exactly as before so they will not be proved as =
they
are simply stated as:<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'margin-left:0cm;mso-add-space:a=
uto;
text-align:justify'><span style=3D'font-size:12.0pt;font-family:"Times New =
Roman"'><o:p>&nbsp;</o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'text-align:justify'><span
style=3D'font-size:12.0pt;font-family:"Times New Roman"'><span style=3D'mso=
-tab-count:
2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y=
<sup>1</sup>
(a) =3D 0 ,<span style=3D'mso-spacerun:yes'>&nbsp; </span>Y(b)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D<span style=3D'mso-spacerun:yes=
'>&nbsp;
</span>0<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'text-align:justify'><span
style=3D'font-size:12.0pt;font-family:"Times New Roman"'><o:p>&nbsp;</o:p><=
/span></p>

<p class=3DListParagraphCxSpMiddle style=3D'text-align:justify'><span
style=3D'font-size:12.0pt;font-family:"Times New Roman"'><span style=3D'mso=
-tab-count:
2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y=
<sup>1</sup>
(b) =3D g(t)=3Dd=3Dconstant<o:p></o:p></span></p>

<p class=3DListParagraphCxSpLast style=3D'text-align:justify'><span
style=3D'font-size:12.0pt;font-family:"Times New Roman"'><span style=3D'mso=
-tab-count:
1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'>Case IV<o:p></o:p></b></p>

<p class=3DListParagraphCxSpFirst align=3Dcenter style=3D'text-align:center=
'><b
style=3D'mso-bidi-font-weight:normal'><span style=3D'font-size:12.0pt;font-=
family:
"Times New Roman"'><o:p>&nbsp;</o:p></span></b></p>

<p class=3DListParagraphCxSpMiddle style=3D'text-align:justify'><span
style=3D'font-size:12.0pt;font-family:"Times New Roman"'><span style=3D'mso=
-tab-count:
2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y=
<sup>1</sup>(a)
=3DY<sup>1</sup>(b)<span style=3D'mso-spacerun:yes'>&nbsp; </span>=3D<span
style=3D'mso-spacerun:yes'>&nbsp; </span>0<o:p></o:p></span></p>

<p class=3DListParagraphCxSpMiddle style=3D'text-align:justify'><span
style=3D'font-size:12.0pt;font-family:"Times New Roman"'><o:p>&nbsp;</o:p><=
/span></p>

<p class=3DListParagraphCxSpLast style=3D'text-align:justify'><span
style=3D'font-size:12.0pt;font-family:"Times New Roman"'><o:p>&nbsp;</o:p><=
/span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><b
style=3D'mso-bidi-font-weight:normal'><span style=3D'font-size:14.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>NEW SOLUTION OF
SCHRODINGER&#8217;S EQUATION</span></b><b style=3D'mso-bidi-font-weight:nor=
mal'><span
style=3D'font-size:14.0pt;font-family:Tahoma'><o:p></o:p></span></b></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><u><span
style=3D'font-size:14.0pt;font-family:Tahoma'><o:p><span style=3D'text-deco=
ration:
 none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><u><span
style=3D'font-family:Tahoma'>Case I<o:p></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal>Y(0)=3DY(1)=3D0</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'>Y&#8217;(0)=3Dconstant<span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>If
Y(0)<span style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0 and Y(1)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>We
obtain from solution of equations <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y
=3D Asin </span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> X + BCos</span><span style=3D'font-size:22.0p=
t'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> x<span style=3D'mso-spacerun:yes'>&nbsp;
</span>--------------- <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&n=
bsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>(1)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sup>1</sup>=3D
B</span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> sin </span><span style=3D'font-size:22.0pt'>&=
#8730;&#955;</span><span
style=3D'font-family:Tahoma'> x<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp=
;&nbsp;
</span>+ A </span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> Cos </span><span style=3D'font-size:22.0pt'>&=
#8730;&#955;</span><span
style=3D'font-family:Tahoma'> x ------ (2)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Putting
in the value x=3D0 in line 1 gives B=3D0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Also
by substituting x=3D1 in Y(1)<span style=3D'mso-spacerun:yes'>&nbsp; </span=
>=3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Gives
<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Y(1)
=3D<span style=3D'mso-spacerun:yes'>&nbsp; </span>Asin </span><span
style=3D'font-size:22.0pt'>&#8730;&#955;</span><span style=3D'font-family:T=
ahoma'>
=3D0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Hence
<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><span style=3D'font-size:22.0pt'>&#8730;&#955;=3Dn&#960;</spa=
n><span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Y<sub>n</sub>
=3D A<sub>n</sub> Sin</span><span style=3D'font-size:22.0pt'>&#8730;&#955;<=
/span><span
style=3D'font-family:Tahoma'>n x <span style=3D'mso-tab-count:1'> </span><s=
pan
style=3D'mso-spacerun:yes'>&nbsp;</span>Y<sup>1</sup><sub>n</sub>(a)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D f<sub>n</sub> (t)<o:p></o:p></=
span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Assuming
f<sub>n</sub> (t) =3D d a constant.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;</span>We have <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>n</sub><sup>1</sup>
=3D A<sub>n</sub></span><span style=3D'font-size:22.0pt'>&#8730;&#955;</spa=
n><span
style=3D'font-family:Tahoma'> Cos </span><span style=3D'font-size:22.0pt'>&=
#8730;&#955;</span><span
style=3D'font-family:Tahoma'> x<span style=3D'mso-tab-count:4'>&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>1<o:p></o:p></sp=
an></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>When
x=3D 0, 1 becomes<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>n</sub>1(0)
=3D A<sub>n </sub></span><span style=3D'font-size:22.0pt'>&#8730;&#955;</sp=
an><span
style=3D'font-family:Tahoma'>=3D d<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>A<sub>n</sub>
<u>=3D d<span style=3D'mso-tab-count:1'>&nbsp; </span></u><span style=3D'ms=
o-tab-count:
6'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>2<o:p></o:p></sp=
an></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span>n</span><span style=3D'font-size:22.0pt'> &#960;</span><span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Y<sub>n</sub>=3D(d/n</span><span style=3D'font-size:22.0pt'>&#960;</=
span><span
style=3D'font-family:Tahoma'>) Sin n</span><span style=3D'font-size:22.0pt'=
> &#960;</span><span
style=3D'font-family:Tahoma'> x <span style=3D'mso-tab-count:1'>&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>n =3D 1,2,3,4,5<span style=3D'mso-t=
ab-count:
1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></spa=
n></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><u><span
style=3D'font-family:Tahoma'>Case II</span></u><span style=3D'font-family:T=
ahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>Y(0)=3DY&#8217;(1)=3D0</p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'>Y&#8217;(0)=3Dconstant<span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>If
Y(0)<span style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0 and Y1(1)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Solution
of equation gives: <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y
=3D A Cos</span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> x + B sin </span><span style=3D'font-size:22.=
0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> X<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sup>1</sup>=3D
A</span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> sin </span><span style=3D'font-size:22.0pt'>&=
#8730;&#955;</span><span
style=3D'font-family:Tahoma'> x + B </span><span style=3D'font-size:22.0pt'=
>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> Cos</span><span style=3D'font-size:22.0pt'>&#=
8730;&#955;</span><span
style=3D'font-family:Tahoma'> x<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Substituting
Y (0) =3D 0<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>Y(1) =
=3D<span
style=3D'mso-spacerun:yes'>&nbsp; </span>0 we obtain<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>0
=3D A Cos0 + B Sin 0<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;
</span>--------------------------- (1)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>0
=3D- </span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> A sin </span><span style=3D'font-size:22.0pt'=
>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> + B </span><span style=3D'font-size:22.0pt'>&=
#8730;&#955;</span><span
style=3D'font-family:Tahoma'> Cos </span><span style=3D'font-size:22.0pt'>&=
#8730;&#955;<span
style=3D'mso-spacerun:yes'>&nbsp; </span>---- (2)</span><span style=3D'font=
-family:
Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>From
1 it is seen that A=3D0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>From
line 2 we get<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>B</span><span
style=3D'font-size:22.0pt'>&#8730;&#955;</span><span style=3D'font-family:T=
ahoma'>Cos</span><span
style=3D'font-size:22.0pt'>&#8730;&#955;</span><span style=3D'font-family:T=
ahoma'>
=3D 0<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><span
style=3D'font-size:22.0pt'>&#8730;&#955;</span><span style=3D'font-family:T=
ahoma'>
=3D (<u>2n+1)</u></span><span style=3D'font-size:22.0pt'> &#960;</span><span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>2<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>n</sub>
=3D B<sub>n</sub> Sin (<u>2n+1</u>) </span><span style=3D'font-size:22.0pt'=
>&#960;</span><span
style=3D'font-family:Tahoma'> x <span style=3D'mso-tab-count:4'>&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>4<o:p></o:p=
></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;
</span><span style=3D'mso-tab-count:4'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;</span>2<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>n</sub><sup>1</sup>(0)
=3D f(t)<span style=3D'mso-spacerun:yes'>&nbsp; </span>=3D d =3D constant<o=
:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Differentiating<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>n</sub><sup>1</sup>
=3D B<sub>n</sub><u>2n+1</u></span><span style=3D'font-size:22.0pt'> &#960;=
</span><span
style=3D'font-family:Tahoma'> Cos (<u>2n+11</u>) </span><span style=3D'font=
-size:
22.0pt'>&#960;</span><span style=3D'font-family:Tahoma'> x <span
style=3D'mso-tab-count:2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>5<o:p></o:p=
></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;
</span>2<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>2<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp; </span>2<=
o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Substituting
Yn1(0) =3D d in line 4<span style=3D'mso-spacerun:yes'>&nbsp; </span>yields=
<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p=
></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>B<sub>n</sub>
=3D<span style=3D'mso-spacerun:yes'>&nbsp; </span><u><span
style=3D'mso-spacerun:yes'>&nbsp;</span>2d<o:p></o:p></u></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>(2n+=
1)</span><span
style=3D'font-size:22.0pt'>&#960;</span><span style=3D'font-family:Tahoma'>=
<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sub>n</sub>
=3D<u> 2d/(2n+1</u>)</span><span style=3D'font-size:22.0pt'> &#960;)</span>=
<u><span
style=3D'font-family:Tahoma'> </span></u><span style=3D'font-family:Tahoma'=
>sin (<u>2n+1</u>)</span><span
style=3D'font-size:22.0pt'> &#960;</span><span style=3D'font-family:Tahoma'=
> x
/2<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>n =
=3D
0,1,2 ------<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><u><span style=3D'font-fam=
ily:Tahoma'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><u><span
style=3D'font-family:Tahoma'>Case III<o:p></o:p></span></u></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal>Y(0)=3DY(1)=3D0</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'>Y&#8217;(1)=3Dconst<span
style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>In
case III, we set Y(x) =3D 0 at x =3D 0 and Y(b) =3D 0 at x =3D 1 in equatio=
ns below <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y
=3D B Cos</span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> x + A sin </span><span style=3D'font-size:22.=
0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> X<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sup>1</sup>=3D
&#8730; &#955; Bsin </span><span style=3D'font-size:22.0pt'>&#8730;&#955;</=
span><span
style=3D'font-family:Tahoma'> x + A</span><span style=3D'font-size:22.0pt'>=
&#8730;&#955;</span><span
style=3D'font-family:Tahoma'>Cos</span><span style=3D'font-size:22.0pt'>&#8=
730;&#955;</span><span
style=3D'font-family:Tahoma'> x<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>gives<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Y<sub>1</sub>(0)
=3D -</span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> A sin 0 + B </span><span style=3D'font-size:2=
2.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> Cos 0 =3D 0<span style=3D'mso-tab-count:2'>&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>B=3D0<o:p></o:p></span>=
</p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>Hence
Y =3D A sin </span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><sp=
an
style=3D'font-family:Tahoma'> x<span style=3D'mso-tab-count:6'>&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp; </span><span
style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>35<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>If
we put Y(x) =3D 0 at x =3D b =3D 1 we obtain<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y
=3D A sin </span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> =3D 0 <span style=3D'mso-tab-count:5'>&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp; </span>36<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:2=
2.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>&#8730;&#955;</span><span style=3D'font-family:Tahoma'> =3D <u>n</u>=
</span><span
style=3D'font-size:22.0pt'>&#960;</span><span style=3D'font-family:Tahoma'>=
<span
style=3D'mso-tab-count:2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>n
=3D 0, 1, 2, ..<span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp; </span>37<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Hence
<span style=3D'mso-tab-count:1'>&nbsp; </span>Y =3D A sin(<u>n</u></span><s=
pan
style=3D'font-size:22.0pt'>&#960;</span><span style=3D'font-family:Tahoma'>=
x)<span
style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp; </span>n
=3D 0,<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span>1, 2, 3, .</span=
><span
style=3D'font-size:22.0pt'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>From
<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y
=3D B sin </span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> x<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sup>1</sup>(b)
=3D g(t) =3D e =3D constant<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>We
obtain by differentially and setting <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>b=3D1
<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y<sup>1</sup>
=3D </span><span style=3D'font-size:22.0pt'>&#8730;&#955;</span><span
style=3D'font-family:Tahoma'> B<sub>2</sub> cos</span><span style=3D'font-s=
ize:
22.0pt'>&#8730;&#955;</span><span style=3D'font-family:Tahoma'> x <span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span=
></span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
 =3D <u>n</u><sup>2</sup></span><span
style=3D'font-size:22.0pt'>&#960;<sup>2</sup></span><span style=3D'font-fam=
ily:
Tahoma'><span style=3D'mso-tab-count:6'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><span style=3D'mso-tab-count:1'> </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>B<sub>n</sub>
=3D 2<u>d</u> <span style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>x
=3D1<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span></span><span
style=3D'font-size:22.0pt'>&#8730;&#955;</span><span style=3D'font-family:T=
ahoma'> <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>=3D<u><span
style=3D'mso-spacerun:yes'>&nbsp; </span>2d<span style=3D'mso-spacerun:yes'=
>&nbsp;
</span></u><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:3'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp; </span>n</span><span
style=3D'font-size:22.0pt'> &#960;</span><span style=3D'font-family:Tahoma'=
><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Y
=3D <u>2dsin</u> <u>n</u></span><span style=3D'font-size:22.0pt'>&#960;</sp=
an><span
style=3D'font-family:Tahoma'> x<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp=
;&nbsp;
</span>n=3D 0,1,2,3<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'mso-tab-cou=
nt:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>n<span
style=3D'font-size:22.0pt'>&#960;</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><b
style=3D'mso-bidi-font-weight:normal'>RESULTS OF SOLUTIONS OF SCHRODINGER&#=
8217;S
EQUATION<o:p></o:p></b></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'>3:4<span style=3D'font-fam=
ily:Tahoma'><o:p></o:p></span></p>

<p class=3Dparagraph><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>I=
t was
shown in the old solution that the energy for the particle is given by k<su=
p>2</sup><sub>1</sub>
=3D (n </span><span lang=3DEN-US style=3D'font-family:Symbol;mso-ansi-langu=
age:EN-US'>p/</span><span
lang=3DEN-US style=3D'mso-ansi-language:EN-US'>L)<sup>2</sup>&nbsp; =3D 2m/=
&#295;<sup>2</sup>
E , or<o:p></o:p></span></p>

<p class=3Dparagraph><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>H=
ere the
interval is chosen as (0, 1) hence L=3D1&nbsp;<o:p></o:p></span></p>

<p class=3Dparagraph align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'><sub><span lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareas=
t-language:
EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"_x0000_i1054" typ=
e=3D"#_x0000_t75"
 style=3D'width:141pt;height:33pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image023.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D188 height=3D44
src=3D"relativistic_files/image023.gif" v:shapes=3D"_x0000_i1054"><![endif]=
></span></sub></b><span
lang=3DEN-US style=3D'mso-ansi-language:EN-US'><o:p></o:p></span></p>

<p class=3Dparagraph align=3Dcenter style=3D'text-align:center'><span
style=3D'font-family:Tahoma'>It can be seen that while old solution</span><=
span
style=3D'mso-ansi-language:EN-US'> <span lang=3DEN-US><o:p></o:p></span></s=
pan></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><i
style=3D'mso-bidi-font-style:normal'><span style=3D'font-family:Tahoma'>TAB=
LE OF
SOLUTIONS<o:p></o:p></span></i></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;
</span><o:p></o:p></span></p>

<table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0
 style=3D'border-collapse:collapse;border:none;mso-border-alt:solid windowt=
ext .5pt;
 mso-yfti-tbllook:480;mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-border-inside=
h:
 .5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;mso-border-alt:so=
lid windowtext .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>CLASS<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-left:none;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>HERMITIAN
  CONDITIONS<o:p></o:p></span></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border:solid windowtex=
t 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>NEW
  SOLUTION<o:p></o:p></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border:solid windowte=
xt 1.0pt;
  border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt:
  solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>ENERGY=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<span
  style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </span><span
  style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span><o:p></o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:1'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-top:none;
  mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext =
.5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>1<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border-top:none;border-left:none;border-bottom:=
solid windowtext 1.0pt;
  border-right:solid windowtext 1.0pt;mso-border-top-alt:solid windowtext .=
5pt;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal>Y(0)=3DY(1)=3D0<span lang=3DEN-US style=3D'mso-ansi-=
language:EN-US;
  mso-fareast-language:EN-US'><o:p></o:p></span></p>
  <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'>Y&#8217;(0)=3Dconstant<s=
pan
  style=3D'font-family:Tahoma'><o:p></o:p></span></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border-top:none;border=
-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p><span lang=3DEN-US>(d/n&#960;)*sin((n&#960;x)</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-fon=
t-weight:
  normal'>En=3D</b><span style=3D'font-size:22.0pt'> (n<sup>2</sup>&#960;<s=
up>2</sup>
  h<sup>2</sup> )/(2m)</span><span style=3D'font-family:Tahoma'><o:p></o:p>=
</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:2'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-top:none;
  mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext =
.5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>2<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border-top:none;border-left:none;border-bottom:=
solid windowtext 1.0pt;
  border-right:solid windowtext 1.0pt;mso-border-top-alt:solid windowtext .=
5pt;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal>Y(0)=3DY&#8217;(1)=3D0<span lang=3DEN-US style=3D'ms=
o-ansi-language:
  EN-US;mso-fareast-language:EN-US'><o:p></o:p></span></p>
  <p class=3DMsoNormal><o:p>&nbsp;</o:p></p>
  <p class=3DMsoNormal>Y&#8217;(0)=3Dconstant<b style=3D'mso-bidi-font-weig=
ht:normal'><span
  lang=3DEN-US style=3D'font-size:36.0pt;mso-ansi-language:EN-US;mso-fareas=
t-language:
  EN-US'><o:p></o:p></span></b></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border-top:none;border=
-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p><span lang=3DEN-US>(d/k&#960;)*sin((k&#960;x)</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>K=3D(2n+1)/2<o:p></o:p></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-fon=
t-weight:
  normal'>En=3D</b><span style=3D'font-size:22.0pt'> (k<sup>2</sup>&#960;<s=
up>2</sup>
  h<sup>2</sup> )/(2m)</span><span style=3D'font-family:Tahoma'><o:p></o:p>=
</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>K=3D(2n+1)/2<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:3'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-top:none;
  mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext =
.5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>3<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border-top:none;border-left:none;border-bottom:=
solid windowtext 1.0pt;
  border-right:solid windowtext 1.0pt;mso-border-top-alt:solid windowtext .=
5pt;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal>Y&#8217;(0)=3DY(1)=3D0<span lang=3DEN-US style=3D'ms=
o-ansi-language:
  EN-US;mso-fareast-language:EN-US'><o:p></o:p></span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><o:p>&nbsp;</o:p></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'>Y&#8217;(1)=3Dconstant<s=
pan
  style=3D'font-family:Tahoma'><o:p></o:p></span></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border-top:none;border=
-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p><span lang=3DEN-US>(d/n&#960;)*sin((n&#960;x)</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-fon=
t-weight:
  normal'>En=3D</b><span style=3D'font-size:22.0pt'> (n<sup>2</sup>&#960;<s=
up>2</sup>
  h<sup>2</sup> )/(2m)</span><span style=3D'font-family:Tahoma'><o:p></o:p>=
</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'><o:p>&nbsp;</o:p></span></p>
  </td>
 </tr>
 <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'>
  <td valign=3Dtop style=3D'border:solid windowtext 1.0pt;border-top:none;
  mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext =
.5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>4<o:p></o:p></span></p>
  </td>
  <td valign=3Dtop style=3D'border-top:none;border-left:none;border-bottom:=
solid windowtext 1.0pt;
  border-right:solid windowtext 1.0pt;mso-border-top-alt:solid windowtext .=
5pt;
  mso-border-left-alt:solid windowtext .5pt;mso-border-alt:solid windowtext=
 .5pt;
  padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal>Y&#8217;(0)=3DY(1)=3D0<span lang=3DEN-US style=3D'ms=
o-ansi-language:
  EN-US;mso-fareast-language:EN-US'><o:p></o:p></span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'>Y&#8217;(1)=3Dcons<span
  style=3D'font-family:Tahoma'><o:p></o:p></span></p>
  </td>
  <td width=3D124 valign=3Dtop style=3D'width:93.2pt;border-top:none;border=
-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p><span lang=3DEN-US>(d/k&#960;)*sin((n&#960;x)</span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>K=3D(2n+1)/2<o:p></o:p></span></p>
  </td>
  <td width=3D236 valign=3Dtop style=3D'width:176.9pt;border-top:none;borde=
r-left:
  none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1=
.0pt;
  mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid window=
text .5pt;
  mso-border-alt:solid windowtext .5pt;padding:0cm 5.4pt 0cm 5.4pt'>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><b style=3D'mso-bidi-fon=
t-weight:
  normal'>En=3D</b><span style=3D'font-size:22.0pt'>((k<sup>2</sup>&#960;<s=
up>2</sup>
  h<sup>2</sup>)/(2m)<o:p></o:p></span></p>
  <p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-fami=
ly:Tahoma'>K=3D(2n+1)/2<o:p></o:p></span></p>
  </td>
 </tr>
</table>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;</span>of Schrodinger&#8217;s equation giv=
es one
wave function the new one gives more than one.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><b
style=3D'mso-bidi-font-weight:normal'><span style=3D'font-size:14.0pt;font-=
family:
Tahoma'>Schrodinger&#8217;s Equation in 3 Dimensions<o:p></o:p></span></b><=
/p>

<p class=3DListParagraph style=3D'mso-margin-top-alt:auto;mso-margin-bottom=
-alt:
auto;margin-left:18.0pt;mso-add-space:auto;text-indent:-18.0pt;mso-list:l1 =
level1 lfo1'><![if !supportLists]><span
style=3D'font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;
mso-bidi-font-family:Symbol'><span style=3D'mso-list:Ignore'>&middot;<span
style=3D'font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span></span></span><![endif]><span style=3D'font-size:14.0pt'>In three
dimensions (Cartesian coordinates) the equation is given as<span
style=3D'mso-spacerun:yes'>&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-lan=
guage:
EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_11"
 o:spid=3D"_x0000_i1055" type=3D"#_x0000_t75" style=3D'width:327pt;height:4=
5pt;
 visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image041.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D436 height=3D60
src=3D"relativistic_files/image041.gif" v:shapes=3D"Picture_x0020_11"><![en=
dif]></span></b></p>

<p class=3DListParagraph style=3D'mso-margin-top-alt:auto;mso-margin-bottom=
-alt:
auto;margin-left:18.0pt;mso-add-space:auto;text-indent:-18.0pt;mso-list:l1 =
level1 lfo1'><![if !supportLists]><span
style=3D'font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;
mso-bidi-font-family:Symbol'><span style=3D'mso-list:Ignore'>&middot;<span
style=3D'font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span></span></span><![endif]><span style=3D'font-size:14.0pt'>If the pote=
ntial
energy separates into the sum of parts <o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-lan=
guage:
EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_12"
 o:spid=3D"_x0000_i1056" type=3D"#_x0000_t75" style=3D'width:186pt;height:1=
8pt;
 visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image042.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D248 height=3D24
src=3D"relativistic_files/image042.gif" v:shapes=3D"Picture_x0020_12"><![en=
dif]></span></b></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><o:p>&nbsp;=
</o:p></p>

<p class=3DListParagraph style=3D'margin-left:18.0pt;mso-add-space:auto;tex=
t-indent:
-18.0pt;mso-list:l1 level1 lfo1'><![if !supportLists]><span style=3D'font-s=
ize:
14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-fami=
ly:
Symbol'><span style=3D'mso-list:Ignore'>&middot;<span style=3D'font:7.0pt "=
Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span></span><![endif]><span style=3D'font-size:14.0pt'>then we can=
 find
solutions that factor into parts: <o:p></o:p></span></p>

<p align=3Dcenter style=3D'text-align:center'><b style=3D'mso-bidi-font-wei=
ght:normal'><span
lang=3DEN-US style=3D'mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"_=
x0000_i1057"
 type=3D"#_x0000_t75" style=3D'width:156pt;height:18pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image043.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D208 height=3D24
src=3D"relativistic_files/image043.gif" v:shapes=3D"_x0000_i1057"><![endif]=
></span></b></p>

<p class=3DListParagraph style=3D'mso-margin-top-alt:auto;mso-margin-bottom=
-alt:
auto;margin-left:18.0pt;mso-add-space:auto;text-indent:-18.0pt;mso-list:l1 =
level1 lfo1'><![if !supportLists]><span
style=3D'font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;
mso-bidi-font-family:Symbol'><span style=3D'mso-list:Ignore'>&middot;<span
style=3D'font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span></span></span><![endif]><span style=3D'font-size:14.0pt'>Substitutin=
g into
the equation we find <o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-lan=
guage:
EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_13"
 o:spid=3D"_x0000_i1058" type=3D"#_x0000_t75" style=3D'width:430.5pt;height=
:46.5pt;
 visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image044.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D574 height=3D62
src=3D"relativistic_files/image044.gif" v:shapes=3D"Picture_x0020_13"><![en=
dif]></span></b></p>

<p class=3DListParagraph style=3D'mso-margin-top-alt:auto;mso-margin-bottom=
-alt:
auto;margin-left:18.0pt;mso-add-space:auto;text-indent:-18.0pt;mso-list:l1 =
level1 lfo1'><![if !supportLists]><span
style=3D'font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;
mso-bidi-font-family:Symbol'><span style=3D'mso-list:Ignore'>&middot;<span
style=3D'font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span></span></span><![endif]><span style=3D'font-size:14.0pt'>By dividing=
 by </span><span
style=3D'font-size:14.0pt;font-family:Symbol'>f</span><sub><span
style=3D'font-size:14.0pt;font-family:Geneva'>x</span></sub><span
style=3D'font-size:14.0pt;font-family:Symbol'>f</span><sub><span
style=3D'font-size:14.0pt;font-family:Geneva'>y</span></sub><span
style=3D'font-size:14.0pt;font-family:Symbol'>f</span><sub><span
style=3D'font-size:14.0pt;font-family:Geneva'>z</span></sub><span
style=3D'font-size:14.0pt;font-family:Geneva'> we have</span><span
style=3D'font-size:14.0pt'> <o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-lan=
guage:
EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_14"
 o:spid=3D"_x0000_i1059" type=3D"#_x0000_t75" style=3D'width:393pt;height:4=
6.5pt;
 visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image045.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D524 height=3D62
src=3D"relativistic_files/image045.gif" v:shapes=3D"Picture_x0020_14"><![en=
dif]></span></b></p>

<p class=3DListParagraph style=3D'mso-margin-top-alt:auto;mso-margin-bottom=
-alt:
auto;margin-left:18.0pt;mso-add-space:auto;text-indent:-18.0pt;mso-list:l1 =
level1 lfo1'><![if !supportLists]><span
style=3D'font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol;
mso-bidi-font-family:Symbol'><span style=3D'mso-list:Ignore'>&middot;<span
style=3D'font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;
</span></span></span><![endif]><span style=3D'font-size:14.0pt'>Assuming th=
at the
wave function is separable we get <o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-lan=
guage:
EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"_x0000_i1060" typ=
e=3D"#_x0000_t75"
 style=3D'width:153pt;height:132pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image046.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D204 height=3D176
src=3D"relativistic_files/image046.gif" v:shapes=3D"_x0000_i1060"><![endif]=
></span></b></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><o:p>&nbsp;=
</o:p></p>

<p class=3DMsoNormal><span style=3D'font-family:Tahoma'>Where <o:p></o:p></=
span></p>

<p align=3Dcenter style=3D'text-align:center'><b style=3D'mso-bidi-font-wei=
ght:normal'><span
lang=3DEN-US style=3D'mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"P=
icture_x0020_15"
 o:spid=3D"_x0000_i1061" type=3D"#_x0000_t75" style=3D'width:97.5pt;height:=
15.75pt;
 visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image047.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D130 height=3D21
src=3D"relativistic_files/image047.gif" v:shapes=3D"Picture_x0020_15"><![en=
dif]></span></b></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><o:p>&nbsp;=
</o:p></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'>If
we assume class 1 hermitian condition the <o:p></o:p></span></p>

<p><span lang=3DEN-US>Y(x)=3D(d/n&#960;)*sin((n&#960;x)</span></p>

<p><span lang=3DEN-US>Y(y)=3D(d/n&#960;)*sin((n&#960;y)</span></p>

<p><span lang=3DEN-US>Y(z)=3D(d/n&#960;)*sin((n&#960;z)</span></p>

<p style=3D'text-align:justify'><span lang=3DEN-US>Hence a three dimensional
solution would be given as</span></p>

<p style=3D'text-align:justify'><span lang=3DEN-US>Y(x)=3D(d/n&#960;)*sin((=
n&#960;x) (d/n&#960;)*sin((n&#960;y)(d/n&#960;)*sin((n&#960;z)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>-----------------<span
style=3D'mso-spacerun:yes'>&nbsp; </span>(1)</span></p>

<p style=3D'text-align:justify'><span lang=3DEN-US>Where for simplicity the=
 n is
kept the same and the constants d is kept the same.</span></p>

<p style=3D'text-align:justify'><span lang=3DEN-US>There are four solutions=
 and
hence there a number of combinations in the three dimensional case. Some may
yield degeneracy but that is not necessary here.</span></p>

<p style=3D'text-align:justify'><span lang=3DEN-US><o:p>&nbsp;</o:p></span>=
</p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;tab-stops:10=
2.0pt'><span
style=3D'font-family:Tahoma'>In the gradient vector form the equation reduc=
es to<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<div align=3Dcenter>

<table class=3DMsoNormalTable border=3D0 cellpadding=3D0 width=3D"100%"
 style=3D'width:100.0%;mso-cellspacing:1.5pt'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'>
  <td nowrap style=3D'padding:.75pt .75pt .75pt .75pt'>
  <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><a name=
=3D"eq:TDSE3D"></a><span
  lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-language:EN-US;
  mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"_x0000_i1062" type=3D=
"#_x0000_t75"
   alt=3D"\begin{displaymath}&#10;i \hbar \frac{\partial \psi({\bf r},t)}{\=
partial t} =3D&#10;- \f...&#10;...r^2}{2m} \nabla^2 \psi({\bf r},t) + V({\b=
f r}) \psi({\bf r},t)&#10;\end{displaymath}"
   style=3D'width:252.75pt;height:36.75pt;visibility:visible'>
   <v:imagedata src=3D"relativistic_files/image048.gif" o:title=3D"end{disp=
laymath}"/>
  </v:shape><![endif]--><![if !vml]><img border=3D0 width=3D337 height=3D49
  src=3D"relativistic_files/image048.gif"
  alt=3D"\begin{displaymath}&#10;i \hbar \frac{\partial \psi({\bf r},t)}{\p=
artial t} =3D&#10;- \f...&#10;...r^2}{2m} \nabla^2 \psi({\bf r},t) + V({\bf=
 r}) \psi({\bf r},t)&#10;\end{displaymath}"
  v:shapes=3D"_x0000_i1062"><![endif]></span></p>
  </td>
  <td width=3D29 style=3D'width:21.5pt;padding:.75pt .75pt .75pt .75pt'>
  <p class=3DMsoNormal align=3Dright style=3D'text-align:right'>(21)</p>
  </td>
 </tr>
</table>

</div>

<p class=3DMsoNormal align=3Dright style=3D'text-align:right'><br clear=3Da=
ll
style=3D'mso-special-character:line-break'>
</p>

<p class=3DMsoNormal>or </p>

<div align=3Dcenter>

<table class=3DMsoNormalTable border=3D0 cellpadding=3D0 width=3D"100%"
 style=3D'width:100.0%;mso-cellspacing:1.5pt'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'>
  <td nowrap style=3D'padding:.75pt .75pt .75pt .75pt'>
  <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><span lan=
g=3DEN-US
  style=3D'mso-ansi-language:EN-US;mso-fareast-language:EN-US;mso-no-proof:=
yes'><!--[if gte vml 1]><v:shape
   id=3D"_x0000_i1063" type=3D"#_x0000_t75" alt=3D"\begin{displaymath}&#10;=
\frac{i \hbar}{f(t)} \frac{df}{dt} =3D \frac{1}{\psi({\bf r})}...&#10;...fr=
ac{\hbar^2}{2m} \nabla^2 + V({\bf r}) \right] \psi({\bf r})&#10;\end{displa=
ymath}"
   style=3D'width:232.5pt;height:41.25pt;visibility:visible'>
   <v:imagedata src=3D"relativistic_files/image049.gif" o:title=3D"end{disp=
laymath}"/>
  </v:shape><![endif]--><![if !vml]><img border=3D0 width=3D310 height=3D55
  src=3D"relativistic_files/image049.gif"
  alt=3D"\begin{displaymath}&#10;\frac{i \hbar}{f(t)} \frac{df}{dt} =3D \fr=
ac{1}{\psi({\bf r})}...&#10;...frac{\hbar^2}{2m} \nabla^2 + V({\bf r}) \rig=
ht] \psi({\bf r})&#10;\end{displaymath}"
  v:shapes=3D"_x0000_i1063"><![endif]></span></p>
  </td>
  <td width=3D10 style=3D'width:7.5pt;padding:.75pt .75pt .75pt .75pt'>
  <p class=3DMsoNormal align=3Dright style=3D'text-align:right'>(23)</p>
  </td>
 </tr>
</table>

</div>

<p class=3DMsoNormal align=3Dright style=3D'text-align:right'><br clear=3Da=
ll
style=3D'mso-special-character:line-break'>
</p>

<p class=3DMsoNormal style=3D'text-align:justify'>Since the left-hand side =
is a
function of <span lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast=
-language:
EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_16"
 o:spid=3D"_x0000_i1064" type=3D"#_x0000_t75" alt=3D"$t$" style=3D'width:8.=
25pt;
 height:14.25pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image050.gif" o:title=3D"$t$"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D11 height=3D19
src=3D"relativistic_files/image050.gif" alt=3D"$t$" v:shapes=3D"Picture_x00=
20_16"><![endif]></span>only
and the right hand side is a function of <span lang=3DEN-US style=3D'mso-an=
si-language:
EN-US;mso-fareast-language:EN-US;mso-no-proof:yes'><!--[if gte vml 1]><v:sh=
ape
 id=3D"Picture_x0020_17" o:spid=3D"_x0000_i1065" type=3D"#_x0000_t75" alt=
=3D"${\bf r}$"
 style=3D'width:9.75pt;height:14.25pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image051.gif" o:title=3D"bf r}$"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D13 height=3D19
src=3D"relativistic_files/image051.gif" alt=3D"${\bf r}$" v:shapes=3D"Pictu=
re_x0020_17"><![endif]></span>only,
the two sides must equal a constant. If we tentatively designate this const=
ant <span
lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-language:EN-US;
mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_18" o:spi=
d=3D"_x0000_i1066"
 type=3D"#_x0000_t75" alt=3D"$E$" style=3D'width:14.25pt;height:12pt;visibi=
lity:visible'>
 <v:imagedata src=3D"relativistic_files/image052.gif" o:title=3D"$E$"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D19 height=3D16
src=3D"relativistic_files/image052.gif" alt=3D"$E$" v:shapes=3D"Picture_x00=
20_18"><![endif]></span>(since
the right-hand side clearly must have the dimensions of energy), then we
extract two ordinary differential equations, namely, </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'text-align:justify'>The latter equation is on=
ce again
the time-independent Schr&ouml;dinger equation. The former equation is easi=
ly
solved to yield,</p>

<p class=3DMsoNormal style=3D'text-align:justify'><o:p>&nbsp;</o:p></p>

<div align=3Dcenter>

<table class=3DMsoNormalTable border=3D0 cellpadding=3D0 width=3D"100%"
 style=3D'width:100.0%;mso-cellspacing:1.5pt'>
 <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'>
  <td nowrap style=3D'padding:.75pt .75pt .75pt .75pt'>
  <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><span lan=
g=3DEN-US
  style=3D'mso-ansi-language:EN-US;mso-fareast-language:EN-US;mso-no-proof:=
yes'><!--[if gte vml 1]><v:shape
   id=3D"Picture_x0020_20" o:spid=3D"_x0000_i1067" type=3D"#_x0000_t75" alt=
=3D"\begin{displaymath}&#10;f(t) =3D e^{-iEt / \hbar}&#10;\end{displaymath}"
   style=3D'width:83.25pt;height:24.75pt;visibility:visible'>
   <v:imagedata src=3D"relativistic_files/image053.gif" o:title=3D"end{disp=
laymath}"/>
  </v:shape><![endif]--><![if !vml]><img border=3D0 width=3D111 height=3D33
  src=3D"relativistic_files/image053.gif"
  alt=3D"\begin{displaymath}&#10;f(t) =3D e^{-iEt / \hbar}&#10;\end{display=
math}"
  v:shapes=3D"Picture_x0020_20"><![endif]></span></p>
  </td>
  <td width=3D10 style=3D'width:7.5pt;padding:.75pt .75pt .75pt .75pt'>
  <p class=3DMsoNormal align=3Dright style=3D'text-align:right'>(26)</p>
  </td>
 </tr>
</table>

</div>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>In our hermition conditions stated before d wa=
s any
constant, some function of time<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>We will arrive at the same conditions if <o:p>=
</o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'><span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;
</span>d=3Df(t)<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-language:EN-US;
mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_21" o:spi=
d=3D"_x0000_i1068"
 type=3D"#_x0000_t75" alt=3D"\begin{displaymath}&#10;f(t) =3D e^{-iEt / \hb=
ar}&#10;\end{displaymath}"
 style=3D'width:83.25pt;height:24.75pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image053.gif" o:title=3D"end{displa=
ymath}"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D111 height=3D33
src=3D"relativistic_files/image053.gif"
alt=3D"\begin{displaymath}&#10;f(t) =3D e^{-iEt / \hbar}&#10;\end{displayma=
th}"
v:shapes=3D"Picture_x0020_21"><![endif]></span><span style=3D'font-family:T=
ahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>Our time independent solution<o:p></o:p></span=
></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'>Y(x)=3D=
(d/n&#960;)*sin((n&#960;x)</p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'>Now bec=
omes </p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'>Y(x,t)=
=3D(1/n&#960;)*sin((n&#960;x)*f(t)<span
style=3D'mso-spacerun:yes'>&nbsp; </span><span style=3D'font-family:Tahoma'=
><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-language:EN-US;
mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_22" o:spi=
d=3D"_x0000_i1069"
 type=3D"#_x0000_t75" alt=3D"\begin{displaymath}&#10;f(t) =3D e^{-iEt / \hb=
ar}&#10;\end{displaymath}"
 style=3D'width:83.25pt;height:24.75pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image053.gif" o:title=3D"end{displa=
ymath}"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D111 height=3D33
src=3D"relativistic_files/image053.gif"
alt=3D"\begin{displaymath}&#10;f(t) =3D e^{-iEt / \hbar}&#10;\end{displayma=
th}"
v:shapes=3D"Picture_x0020_22"><![endif]></span><span style=3D'font-family:T=
ahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>Where E=3DE1, the component of the energy alon=
g the
x-axis<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'>Y(x, y,=
 z,
t)=3D(1/n&#960;)*sin((n&#960;x) (1/n&#960;)*sin((n&#960;y)(1/n&#960;)*sin((=
n&#960;z)*f(t)<span
style=3D'mso-spacerun:yes'>&nbsp; </span>-----------------<span
style=3D'mso-spacerun:yes'>&nbsp; </span><span style=3D'font-family:Tahoma'=
><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>Where f(t) is given as<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
lang=3DEN-US style=3D'mso-ansi-language:EN-US;mso-fareast-language:EN-US;
mso-no-proof:yes'><!--[if gte vml 1]><v:shape id=3D"Picture_x0020_23" o:spi=
d=3D"_x0000_i1070"
 type=3D"#_x0000_t75" alt=3D"\begin{displaymath}&#10;f(t) =3D e^{-iEt / \hb=
ar}&#10;\end{displaymath}"
 style=3D'width:83.25pt;height:24.75pt;visibility:visible'>
 <v:imagedata src=3D"relativistic_files/image053.gif" o:title=3D"end{displa=
ymath}"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D111 height=3D33
src=3D"relativistic_files/image053.gif"
alt=3D"\begin{displaymath}&#10;f(t) =3D e^{-iEt / \hbar}&#10;\end{displayma=
th}"
v:shapes=3D"Picture_x0020_23"><![endif]></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'>And E i=
s the
total energy where<span style=3D'font-family:Tahoma'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>E=3DE1+E2+E3<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'>Where E1, E2, E3 the components of the energy =
along
the x, y, and z axis respectively.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-align:justify;tab-stops:102.0pt'><span
style=3D'font-family:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-family=
:Tahoma'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

</div>

</body>

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