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logy<o:p></o:p></span></p>
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=3DSpellE>Brockley</span>
      Road</st1:address></st1:Street><o:p></o:p></span></p>
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bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp=
;&nbsp;&nbsp;
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oman</span>
    381/20 <st1:place w:st=3D"on"><st1:country-region w:st=3D"on">Ghana</st=
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    <o:p></o:p></span></p>
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nbsp;
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</span></span><![endif]><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:16.0pt'><o:p></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>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:16.0pt'>Title: Relativistic Proof Electron Has Two Waves=
.<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 <span class=3DSpellE>Kwaku</span> <span class=3DSpellE>=
Boateng</span></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>

<div style=3D'mso-element:para-border-div;border:none;border-bottom:double =
windowtext 2.25pt;
padding:0cm 0cm 1.0pt 0cm'>

<p class=3DMsoNormal style=3D'border:none;mso-border-bottom-alt:double wind=
owtext 2.25pt;
padding:0cm;mso-padding-alt:0cm 0cm 1.0pt 0cm'><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><span
style=3D'mso-spacerun:yes'>&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'border:none;mso-border-bottom-alt:double wind=
owtext 2.25pt;
padding:0cm;mso-padding-alt:0cm 0cm 1.0pt 0cm'><o:p>&nbsp;</o:p></p>

</div>

<p class=3DMsoNormal style=3D'text-align:justify'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'text-align:justify'>UNDER CONSTRUCTION spread=
</p>

<p class=3DMsoNormal style=3D'text-align:justify'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'text-align:justify'>=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 style=3D'text-align:justify'>Solution of Relativistic
Klein-Gordon Schrodinger equation Fig 1.0.2 has shown that the electron has=
 two
waves fig 1.0.3. <span class=3DGramE>(K Wave and B Wave).</span> These two =
waves
are diverging and converging at their centre. This indicates that the elect=
ron
is a dipole.</p>

<p class=3DMsoNormal style=3D'text-align:justify'>=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 style=3D'text-align:justify'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;</span>Schrodinger gave the eq=
uation
below Fig 1.0.1</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"/>
   <v:formulas>
    <v:f eqn=3D"if lineDrawn pixelLineWidth 0"/>
    <v:f eqn=3D"sum @0 1 0"/>
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    <v:f eqn=3D"prod @3 21600 pixelWidth"/>
    <v:f eqn=3D"prod @3 21600 pixelHeight"/>
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    <v:f eqn=3D"sum @8 21600 0"/>
    <v:f eqn=3D"prod @7 21600 pixelHeight"/>
    <v:f eqn=3D"sum @10 21600 0"/>
   </v:formulas>
   <v:path o:extrusionok=3D"f" gradientshapeok=3D"t" o:connecttype=3D"rect"=
/>
   <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"spread_files/image001.jpg" o:href=3D"mhtml:file://F:=
\universal_websites\word_pages_=3D20062008\wordintroductiontoquantummechani=
cs.mht!http://www.physlink.com/Education/AskExperts/Images/ae329a.jpg"/>
  </v:shape><![endif]--><![if !vml]><img border=3D0 width=3D494 height=3D159
  src=3D"spread_files/image002.jpg" v:shapes=3D"_x0000_i1025"><![endif]></s=
pan></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;
</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;</span=
>Fig
1.0.1</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-<span class=3DSpellE>Gorden</span> 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"spread_files/image003.gif" o:title=3D","/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D148 height=3D22
src=3D"spread_files/image003.gif" alt=3D"E^2 =3D p^2c^2 + m^2c^4\," v:shape=
s=3D"Picture_x0020_1"><![endif]></span></p>

<p><span class=3DGramE><span lang=3DEN-US>to</span></span><span lang=3DEN-U=
S> 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"spread_files/image004.gif" o:title=3D","/>
  </v:shape><![endif]--><![if !vml]><img border=3D0 width=3D240 height=3D44
  src=3D"spread_files/image004.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><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>FIG 1.0 .2=
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>which</span> is <span class=3DSpel=
lE>relativistically</span>
invariant.</p>

<p><span lang=3DEN-US>As before we replace the Greek letter by Y(<span
class=3DSpellE>x<span class=3DGramE>,t</span></span>).</span></p>

<p><span lang=3DEN-US>Also assuming that x, and t are independent and also =
Y(<span
class=3DSpellE>x<span class=3DGramE>,t</span></span>) <span
style=3D'mso-spacerun:yes'>&nbsp;</span>are <span class=3DSpellE>seperable<=
/span></span></p>

<p><span lang=3DEN-US>Y(<span class=3DSpellE>x<span class=3DGramE>,t</span>=
</span>)=3DX(x)T(t)</span></p>

<p><span lang=3DEN-US>Where X is space variable and T is time<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span></span></p>

<p><!--[if gte vml 1]><v:shape id=3D"_x0000_s1036" type=3D"#_x0000_t202" st=
yle=3D'position:absolute;
 margin-left:90pt;margin-top:-54pt;width:260.4pt;height:422.7pt;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;positi=
on:
relative;z-index:11'><span style=3D'position:absolute;left:119px;top:-73px;
width:353px;height:570px'>

<table cellpadding=3D0 cellspacing=3D0>
 <tr>
  <td width=3D353 height=3D570 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_s1036" 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:245.25pt;height:414.75pt'>
     <v:imagedata src=3D"spread_files/image005.jpg" o:title=3D"scan0003"/>
    </v:shape><![endif]--><![if !vml]><img width=3D327 height=3D553
    src=3D"spread_files/image006.jpg" v:shapes=3D"_x0000_i1030"><![endif]><=
/p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  </span><![endif]><![if !mso & !vml]>&nbsp;<![endif]><![if !vml]></td>
 </tr>
</table>

</span></span><![endif]><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<br style=3D'mso-ignore:vglayout' clear=3DALL>

<p><span lang=3DEN-US>K Wave</span></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US>B Wave</span></p>

<p><span lang=3DEN-US><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></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US><o:p>&nbsp;</o:p></span></p>

<p><span lang=3DEN-US><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;&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;</span>FIG
1.0.3Also 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'><span
style=3D'mso-spacerun:yes'>&nbsp;</span><span class=3DSpellE><span class=3D=
GramE>T<sub>tt</sub></span></span></span><span
class=3DGramE><span lang=3DEN-US><span style=3D'mso-spacerun:yes'>&nbsp; </=
span>as</span></span><span
lang=3DEN-US> 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
class=3DGramE><span style=3D'font-size:16.0pt'>X<sub>xx<span
style=3D'mso-spacerun:yes'>&nbsp; </span>+</sub></span></span><sub><span
style=3D'font-size:16.0pt'> </span></sub><span style=3D'font-size:16.0pt'><=
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 class=3DSpellE><span style=3D'font-size:16.0pt'>=
T<sub>tt</sub></span></span><sub><span
style=3D'font-size:16.0pt'> + </span></sub><span style=3D'font-size:16.0pt'=
>k<sup>2</sup>
<span class=3DSpellE><span class=3DGramE>T<sub>tt</sub></span></span><span
class=3DGramE><sub><span style=3D'mso-spacerun:yes'>&nbsp; </span></sub>=3D=
</span>0</span><sub><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> <span class=3DGramE>h<sup>2 </sup><span style=3D'font-size:1=
2.0pt'>)</span></span></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
&plusmn; <span class=3DGramE>&#8730;(</span><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/</su=
p> 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>L=3D <span class=3DSpellE><span style=3D'font-size:14.=
0pt'>m&#955;</span></span><span
style=3D'font-size:14.0pt'><span style=3D'mso-spacerun:yes'>&nbsp; </span><=
span
style=3D'mso-spacerun:yes'>&nbsp;</span>m=3D1, 2, 3, 4, 5 ----------</span>=
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>which</span> are multiples of wave=
lengths
of the waves. At any point the time is distance divided by the speed of lig=
ht.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>t=3D <span class=3DSpellE>m<span style=3D'font-size:14=
.0pt'>&#955;</span></span>/c<span
style=3D'mso-spacerun:yes'>&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;</span>m=3D1, 2, 3, 4, 5--------</p>

<p class=3DMsoNormal><span class=3DGramE>where</span> c is the speed of lig=
ht.</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><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE><span style=3D'font-size:16.0pt'>T=
(</span></span><span
style=3D'font-size:16.0pt'>t)=3DA. sin(<span class=3DSpellE>kt</span>)<o:p>=
</o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>The <span class=3DGramE>X<span style=3D'mso-spacerun:y=
es'>&nbsp;
</span>part</span> 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 lfo1'><![if !supportLists]><span
style=3D'mso-list:Ignore'>(1)<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;&nbsp;
</span></span><![endif]><span class=3DSpellE>Dirichlet&#8217;s</span> bound=
ary
conditions</p>

<p class=3DMsoNormal style=3D'margin-left:18.0pt;text-indent:-18.0pt;mso-li=
st:l0 level1 lfo1'><![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 lfo1'><![if !supportLists]><span
style=3D'mso-list:Ignore'>(3)<span style=3D'font:7.0pt "Times New Roman"'>&=
nbsp;&nbsp;
</span></span><![endif]><span class=3DSpellE>Hermitian</span> 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'><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>Table 1<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 class=3DSpellE><span style=3D'font-size:18.0pt=
'>Dirichlet&#8217;s</span></span><span
  style=3D'font-size:18.0pt'> Boundary 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 class=3DSpellE><span style=3D'font-size:18.0pt=
'>Hermitian</span></span><span
  style=3D'font-size:18.0pt'> 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 <span
class=3DSpellE>Hermitian</span> conditions for the existence of solution wi=
th
real values of energy as<o:p></o:p></span></p>

<p class=3DMsoNormal><span class=3DGramE><span style=3D'font-size:18.0pt'>X=
(</span></span><span
style=3D'font-size:18.0pt'>0)=3DX(a)=3D0 <o:p></o:p></span></p>

<p class=3DMsoNormal><span class=3DGramE><span style=3D'font-size:18.0pt'>X=
&#8217;(</span></span><span
style=3D'font-size:18.0pt'>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><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></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_s1026" 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_s1026" 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"spread_files/image007.png" o:title=3D"graph fig 1 =
or 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"spread_files/image008.gif" v:shapes=3D"_x0000_i1031"><![endif]><=
/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 conditions as given in table 1.<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 <span class=3DSpellE><span class=3DGramE>Asin</span></span></span><span
class=3DGramE><span style=3D'font-size:22.0pt'>(</span></span><span
style=3D'font-size:22.0pt'>&#955;</span><span style=3D'font-family:Tahoma'>=
 X) + <span
class=3DSpellE>BCos</span>(</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
<span class=3DSpellE>B<span style=3D'font-size:22.0pt;font-family:"Times Ne=
w Roman"'>&#955;</span></span>
<span class=3DGramE>sin(</span></span><span style=3D'font-size:22.0pt'>&#95=
5;</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 <span class=3DGramE>X(</span>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><span
class=3DGramE>X(</span>1) =3D<span style=3D'mso-spacerun:yes'>&nbsp; </span=
><span
class=3DSpellE>Asin<span style=3D'font-size:22.0pt;font-family:"Times New R=
oman"'>&#955;</span></span>
=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'> <=
span
class=3DSpellE>n&#960;</span></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><span class=3DSp=
ellE>X<sub>n</sub></span>
=3D A<sub>n</sub> *<span class=3DGramE>sin<span style=3D'font-size:22.0pt;f=
ont-family:
"Times New Roman"'>(</span></span></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;
</span><span style=3D'mso-spacerun:yes'>&nbsp;</span>X<sup>1</sup><sub>n</s=
ub>(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 d is 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 <span
class=3DSpellE>A<sub>n</sub><span style=3D'font-size:22.0pt;font-family:"Ti=
mes New Roman"'>&#955;</span></span>
Cos </span><span style=3D'font-size:22.0pt'>(&#955;</span><span style=3D'fo=
nt-family:
Tahoma'> x)<span style=3D'mso-tab-count:2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&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><span class=3DSpellE><=
span
class=3DGramE>X<sub>n</sub></span></span><span class=3DGramE>&#8217;(</span=
>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'>&n=
bsp;
</span><span style=3D'mso-spacerun:yes'>&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 class=3DGramE>/(</s=
pan></span><span
class=3DSpellE><span style=3D'font-size:16.0pt'>n&#960;</span></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><span class=3DSpellE>X<sub>n</sub><=
/span>=3D d<span
class=3DGramE>/(</span></span><span style=3D'font-size:22.0pt'> </span><span
class=3DSpellE><span style=3D'font-size:16.0pt'>n&#960;</span></span><span
style=3D'font-family:Tahoma'>) Sin(</span><span class=3DSpellE><span
style=3D'font-size:16.0pt'>n&#960;</span><span style=3D'font-family:Tahoma'=
>x</span></span><span
style=3D'font-family:Tahoma'>) <span style=3D'mso-tab-count:1'>&nbsp; </spa=
n><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-family:Tahoma'><o:p>&nbsp;</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_s1028" 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_s1028" 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"spread_files/image009.png" o:title=3D"relativistic=
 no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"spread_files/image010.gif" v:shapes=3D"_x0000_i1032"><![endif]><=
/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'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span 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;
</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
</span>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><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'><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'><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'>Shape 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, the same energy,
frequency and mass.<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><!--[if gte vml 1]><v:shape id=3D"_x0000_s1027" 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_s1027" 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"spread_files/image011.png" o:title=3D"relativistic=
 spread no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"spread_files/image012.gif" v:shapes=3D"_x0000_i1033"><![endif]><=
/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;
</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;</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;
</span>=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 2<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 2<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 class=3DSpellE><span style=3D'font-size:18.0pt=
'>Dirichlet&#8217;s</span></span><span
  style=3D'font-size:18.0pt'> Boundary 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 <sp=
an
  class=3DSpellE>onditions</span><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 class=3DSpellE><span style=3D'font-size:18.0pt=
'>Hermitian</span></span><span
  style=3D'font-size:18.0pt'> 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'><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 <span
class=3DSpellE>Hermitian</span> conditions for the existence of solution wi=
th
real values of energy are <span class=3DGramE>X(</span>0)=3D0 X&#8217;(0)=
=3Dd <o:p></o:p></span></p>

<p class=3DMsoNormal><span class=3DGramE><span style=3D'font-size:18.0pt'>X=
(</span></span><span
style=3D'font-size:18.0pt'>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 must pass 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 class=3DGramE><span style=3D'font-size:18.0pt'>X=
&#8217;(</span></span><span
style=3D'font-size:18.0pt'>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><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_s1029" 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_s1029" 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"spread_files/image013.png" o:title=3D"graph  fig 3=
"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"spread_files/image014.gif" v:shapes=3D"_x0000_i1034"><![endif]><=
/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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.2.1<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;
</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'><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:14.0pt'>In this case the cond=
itions
of <span class=3DSpellE>Dirichlet</span> boundary and Neumann initial condi=
tions
are as follows:<o:p></o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>X(</span>0)=3DX&#8217;(1)=3D0</p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span class=3DGramE>X&#821=
7;(</span>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
<span class=3DGramE>X(</span>0)<span style=3D'mso-spacerun:yes'>&nbsp; </sp=
an>=3D 0
and X1(1)<span style=3D'mso-spacerun:yes'>&nbsp; </span>=3D 0<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'>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 <span class=3DSpellE>Cos<span style=3D'font-size:22.0pt;font-family:"=
Times New Roman"'>&#955;</span></span>
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
<span class=3DSpellE>A<span style=3D'font-size:22.0pt;font-family:"Times Ne=
w Roman"'>&#955;</span></span>
sin </span><span style=3D'font-size:22.0pt'>&#955;</span><span style=3D'fon=
t-family:
Tahoma'> x + B </span><span style=3D'font-size:22.0pt'>&#955;</span><span
style=3D'font-family:Tahoma'> <span class=3DSpellE>Cos<span style=3D'font-s=
ize:22.0pt;
font-family:"Times New Roman"'>&#955;</span></span> 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><span
class=3DGramE>X&#8217;(</span>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'>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 class=3DGramE><span
style=3D'font-size:22.0pt'>&#955;<span style=3D'mso-spacerun:yes'>&nbsp;
</span>----</span></span><span style=3D'font-size:22.0pt'> (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 class=3DSpellE><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'>&#9=
55;</span></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 class=3DSpellE><span
style=3D'font-family:Tahoma'>X<sub>n</sub></span></span><span style=3D'font=
-family:
Tahoma'> =3D <span class=3DSpellE>B<sub>n</sub></span> Sin (<u>2n+1</u>) </=
span><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'>x <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; </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 class=3DGramE><span
style=3D'font-family:Tahoma'>X<sub>n</sub><sup>1</sup>(</span></span><span
style=3D'font-family:Tahoma'>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
<span class=3DGramE>Xn1(</span>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 class=3DSpellE><span
style=3D'font-family:Tahoma'>B<sub>n</sub></span></span><span style=3D'font=
-family:
Tahoma'> =3D<span style=3D'mso-spacerun:yes'>&nbsp; </span><u><span
style=3D'mso-spacerun:yes'>&nbsp;</span>2d<span class=3DGramE>/<span
style=3D'text-decoration:none;text-underline:none'>(</span></span></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 class=3DSpellE><span
style=3D'font-family:Tahoma'>X<sub>n</sub></span></span><span style=3D'font=
-family:
Tahoma'> =3D<u> 2d<span class=3DGramE>/(</span>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-si=
ze:22.0pt'>
</span><span class=3DSpellE><span style=3D'font-size:16.0pt'>&#960;</span><=
span
style=3D'font-family:Tahoma'>x</span></span><span style=3D'font-family:Taho=
ma'>
/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 1.2.2<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><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></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_s1030" 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_s1030" 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"spread_files/image015.png" o:title=3D"relativistic=
 no 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"spread_files/image016.gif" v:shapes=3D"_x0000_i1035"><![endif]><=
/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;
</span><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'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span 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 1.2.2<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'>Shape 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, the same energy,
frequency and mass.<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><!--[if gte vml 1]><v:shape id=3D"_x0000_s1031" 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_s1031" 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"spread_files/image017.png" o:title=3D"relativistic=
 spread no 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"spread_files/image018.gif" v:shapes=3D"_x0000_i1036"><![endif]><=
/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'>Fig 1.2.3<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'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span 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=3D=3D<span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;
</span>Summary of work<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
the graph fig 1.5.1 below it can be seen that the particle or electron has =
two
waves. This gives only a simplified version of the two adjacent waves.</spa=
n><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><!--[if gte vml 1]><v:shape id=3D"_x0000_s1034" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:117pt;margin-top:20pt;width:251.4pt;
 height:289.6pt;z-index:9;mso-wrap-style:none'/><![endif]--><![if !vml]><sp=
an
style=3D'mso-ignore:vglayout'>

<table cellpadding=3D0 cellspacing=3D0 align=3Dleft>
 <tr>
  <td width=3D155 height=3D26></td>
 </tr>
 <tr>
  <td></td>
  <td width=3D341 height=3D392 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_s1034" 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"spread_files/image011.png" o:title=3D"relativistic=
 spread no 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"spread_files/image012.gif" v:shapes=3D"_x0000_i1028"><![endif]><=
/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>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&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; </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;</s=
pan><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_s1035" 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_s1035" 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"spread_files/image017.png" o:title=3D"relativistic=
 spread no 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"spread_files/image019.gif" v:shapes=3D"_x0000_i1029"><![endif]><=
/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;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;
</span>Fig 1.2.4<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp; </span>=
Electron
consists of two waves.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:1=
8.0pt'>Since
obviously more research has to go on to find the behaviour of these two wav=
es,
more will be said on this in the next papers.<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'tab-stops:102.0pt'><span style=3D'font-size:1=
8.0pt'>Thank
you.<o:p></o:p></span></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

</div>

</body>

</html>

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