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bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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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: Non- Probabilistic Solution of
Schrodinger&#8217;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; </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;&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>Email:
<a href=3D"mailto:info@newquantummechanics.com">info@newquantummechanics.co=
m</a></p>

<p class=3DMsoNormal><u><span style=3D'font-size:16.0pt'><o:p><span
 style=3D'text-decoration:none'>&nbsp;</span></o:p></span></u></p>

<p class=3DMsoNormal><u><span style=3D'font-size:16.0pt'>Scientific Revolut=
ion<o:p></o:p></span></u></p>

<p class=3DMsoNormal>In Quantum Mechanics so much depends upon the probabil=
ity
concept in physics<span style=3D'mso-spacerun:yes'>&nbsp; </span>e g cosmol=
ogy,
atomic physics, nuclear engineering, atom smashing. A new solution has emer=
ged.<u><span
style=3D'font-size:18.0pt'><o:p></o:p></span></u></p>

<p class=3DMsoNormal>The equation given in fig 1.0 was derived by Schroding=
er.
Neil Bohr, and Max Born gave probability solution backed by <st1:place w:st=
=3D"on"><st1:City
 w:st=3D"on">Copenhagen</st1:City></st1:place> interpretation. This new sol=
ution
seems to cast new light onto the solution without probability.</p>

<p class=3DMsoNormal><b style=3D'mso-bidi-font-weight:normal'><u><span
style=3D'font-size:16.0pt'>EPR of 1935</span></u></b></p>

<p class=3DMsoNormal style=3D'text-align:justify'><span
style=3D'mso-spacerun:yes'>&nbsp;</span>Einstein, Polosky, and Erwin wrote a
paper in 1935 to condemn this probability concept in physics. Schrodinger
himself did not accept this ide<b style=3D'mso-bidi-font-weight:normal'><sp=
an
style=3D'font-size:16.0pt'>a<o:p></o:p></span></b></p>

<p class=3DMsoNormal style=3D'text-align:justify'><b style=3D'mso-bidi-font=
-weight:
normal'><span style=3D'font-size:16.0pt'>New Solution<o:p></o:p></span></b>=
</p>

<p class=3DMsoNormal style=3D'text-align:justify'>The new solution given by
Professor Kwaku Boateng has many benefits in cosmology, big bang, black hol=
e,
string theory, atom smashing, paranormal studies etc. </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'text-align:justify;border:none;mso-border-bot=
tom-alt:
double windowtext 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>

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    <v:f eqn=3D"prod @2 1 2"/>
    <v:f eqn=3D"prod @3 21600 pixelWidth"/>
    <v:f eqn=3D"prod @3 21600 pixelHeight"/>
    <v:f eqn=3D"sum @0 0 1"/>
    <v:f eqn=3D"prod @6 1 2"/>
    <v:f eqn=3D"prod @7 21600 pixelWidth"/>
    <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"nonprobabilistic_files/image002.jpg" o:href=3D"mhtml=
:file://F:\universal_websites\word_pages_=3D20062008\wordintroductiontoquan=
tummechanics.mht!http://www.physlink.com/Education/AskExperts/Images/ae329a=
.jpg"/>
  </v:shape><![endif]--><![if !vml]><img border=3D0 width=3D494 height=3D159
  src=3D"nonprobabilistic_files/image003.jpg" v:shapes=3D"_x0000_i1025"><![=
endif]></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><b style=3D'mso-bidi-font-weight:normal'><span
style=3D'font-size:20.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span>Fig 1.0<o:p></o:p></span></b></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>The equation above is given in Greek alphabet but for
simplicity we shall replace this with Y.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></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]>Dirichlet&#8217;s boundary 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]>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_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"nonprobabilistic_files/image004.png" o:title=3D"gr=
aph fig 1 or 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image005.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><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 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 the general Schrodinger equation is given as<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(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;&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;(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'><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;&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>Y<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'><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'>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_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_i1030" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image006.png" o:title=3D"gr=
aph fig 5 non relativistic solution"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image007.gif" 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><![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;
</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
1.1.2<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..1=
.3 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><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_i1029" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image008.png" o:title=3D"gr=
aph spread example"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image009.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'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span 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, 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_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_i1032" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image004.png" o:title=3D"gr=
aph fig 1 or 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_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; </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;</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;&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 1.2.1<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 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 the general Schrodinger equation is given as<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(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;&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: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;&nbsp;&nbsp; </span><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'><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'>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_s1032" 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_s1032" 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"nonprobabilistic_files/image006.png" o:title=3D"gr=
aph fig 5 non relativistic solution"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image007.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'><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 values of d=3D2 an=
d d=3D3
at the same energy, frequency and mass. This means that the particle spread=
s 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><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></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_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_i1034" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image008.png" o:title=3D"gr=
aph spread example"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image011.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;
</span>Fig 1.2.3<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;&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'>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'>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
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><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_s1033" 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_s1033" 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"nonprobabilistic_files/image012.png" o:title=3D"gr=
aph  fig 3"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image013.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;&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;
</span><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span>Fig
1.3.1<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;</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 solution unde=
r 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
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'>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_s1034" 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_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_i1036" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image014.png" o:title=3D"gr=
aph second wave non relativistic"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image015.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'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span 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_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_i1037" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image016.png" o:title=3D"gr=
aph spread of electron second wave"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image017.gif" v:shapes=3D"_x0000_i1037"><=
![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'>Fig 1.3.3<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'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span style=3D'font-size:18.0pt'><o:p>&nbsp;</o:p></sp=
an></p>

<p class=3DMsoNormal><span 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><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_i1038" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image018.png" o:title=3D"gr=
aph fig 4"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image019.gif" v:shapes=3D"_x0000_i1038"><=
![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>

<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'>Fig 1.4.1<span
style=3D'mso-spacerun:yes'>&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;</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:14.0pt'>We seek solution unde=
r the
conditions<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;=
&nbsp;&nbsp;&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: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><!--[if gte vml 1]><v:shape id=3D"_x0000_s1040" 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_s1040" 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"nonprobabilistic_files/image006.png" o:title=3D"gr=
aph fig 5 non relativistic solution"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image020.gif" v:shapes=3D"_x0000_i1039"><=
![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'><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'><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..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><!--[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:16;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:16'>
  <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_i1040" ty=
pe=3D"#_x0000_t75"
     style=3D'width:236.25pt;height:236.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image008.png" o:title=3D"gr=
aph spread example"/>
    </v:shape><![endif]--><![if !vml]><img width=3D315 height=3D315
    src=3D"nonprobabilistic_files/image009.gif" v:shapes=3D"_x0000_i1040"><=
![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'>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_=
i1026"
 type=3D"#_x0000_t75" style=3D'width:141pt;height:33pt;visibility:visible'>
 <v:imagedata src=3D"nonprobabilistic_files/image021.gif" o:title=3D""/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D188 height=3D44
src=3D"nonprobabilistic_files/image021.gif" v:shapes=3D"Picture_x0020_5"><!=
[endif]></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'font-family:Tahoma'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>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></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'><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>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&#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><o:p>&nbsp;</o:p></p>
  <p class=3DMsoNormal>Y&#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>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 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/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>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'>((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'>.
Y<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'><span style=3D'font-family=
:Tahoma'>See
graphs fig 1.4.4 and fig 1.4.5 below which show two waves which are ninty
degrees out of phase.</span><span style=3D'font-size:18.0pt'><span
style=3D'mso-spacerun:yes'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></sp=
an><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-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'><o:p>&nbsp;</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'><o:p>&nbsp;</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_s1037" type=
=3D"#_x0000_t202"
 style=3D'position:absolute;margin-left:-36pt;margin-top:13.05pt;width:242.=
4pt;
 height:235.2pt;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;positi=
on:
absolute;z-index:12;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:12'>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div v:shape=3D"_x0000_s1037" 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_i1027" ty=
pe=3D"#_x0000_t75"
     style=3D'width:227.25pt;height:227.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image022.png" o:title=3D"gr=
aph n equals 1"/>
    </v:shape><![endif]--><![if !vml]><img width=3D303 height=3D303
    src=3D"nonprobabilistic_files/image023.gif" v:shapes=3D"_x0000_i1027"><=
![endif]></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_s1036" type=3D"#_=
x0000_t202"
 style=3D'position:absolute;margin-left:225pt;margin-top:8.35pt;width:233.4=
pt;
 height:226.2pt;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;positi=
on:
absolute;z-index:11;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: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_i1028" ty=
pe=3D"#_x0000_t75"
     style=3D'width:218.25pt;height:218.25pt'>
     <v:imagedata src=3D"nonprobabilistic_files/image024.png" o:title=3D"gr=
aph n equals 2"/>
    </v:shape><![endif]--><![if !vml]><img width=3D291 height=3D291
    src=3D"nonprobabilistic_files/image025.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'><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'><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;
</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;
</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
1.4.5<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><o:p>&nbsp;</o:p></p>

</div>

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