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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "slosh.mws" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "7/15/96   solve for a p
article sloshing around in a double potential well" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "This is a program to fin
d the lowest solutions of a double well problem so that an animation c
an be done of the " }}{PARA 0 "" 0 "" {TEXT -1 67 "electron tunneling \+
back and forth between the sides of the barrier." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "The 'radius' is the valu
e where we run into the infinite well wall. The width is the barrier w
idth and the height is that " }}{PARA 0 "" 0 "" {TEXT -1 15 "of the ba
rrier." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "restart;with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 57 "const:=(6.626E-34*1E9/(2*Pi))^2 /(2*9.1E-31 * 1.602E-19);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ck:=37645/(100000*Pi^2);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "V := a*Heaviside(x-b);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Decimals:=15;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "width:=5; height:=1; radius:= 6;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 69 "plot(subs(a=height,b=width,V),x=0..radius
); # half of the double well" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 65 "eq1:=-ck*diff(y(x),x,x) +subs(a=height,b=width,V)*y(x) = eV*y(x)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
87 "The origin of coordinates for the following calcs is in the corner
 of an infinite well." }}{PARA 0 "" 0 "" {TEXT -1 41 "The wave functio
n starts off as sin (kx)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 69 "for j from 0.0139445*height to 0.013945*heig
ht by 0.0000001*height do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k:=sqr
t(j/ck): j:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "Y_even:=dsolve(\{sub
s(eV=j,eq1),D(y)(width)=k*cos(k*width),y(width)=sin(k*width)\},y(x),nu
meric): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "rhs(Y_even(radius)[2]);
 od;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 89 "The following lines of code were revised repeatedly to obtain a
 solution close to y=0 at " }}{PARA 0 "" 0 "" {TEXT -1 26 "the center \+
of the barrier." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Energy:=13944793
*height/1000000000; k:=sqrt(Energy/ck); # solve for y=0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 " Y_even:=dsolve(\{subs(eV=Energy,e
q1),D(y)(width)=k*cos(k*width),y(width)=sin(k*width)\},y(x),numeric): \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " odeplot(Y_even,[x,y(x)
],0..radius);Y_even(radius);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 119 "The following lines of code were revis
ed repeatedly to obtain a solution close to dy/dx=0 at the center of t
he barrier." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Energy1:=139446326*h
eight/10000000000; k1:=sqrt(Energy1/ck); # solve for dydx=0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "Y_even:=dsolve(\{subs(eV=En
ergy1,eq1),D(y)(width)=k1*cos(k1*width),y(width)=sin(k1*width)\},y(x),
numeric): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "odeplot(Y_eve
n,[x,y(x)],0..radius);Y_even(radius);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " x:=Energy-Energy1;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " freq:=x*1.602E-19/6.63
6E-34; # difference between frequencies" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "get wave function for sym
metric mode:  cos (kx + c), cosh kappa x,  kR+c = Pi/2" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "kappa1:=sqrt((
height-Energy1)/ck);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eq1
:= cos(Pi/2 - k1*width) = B*cosh(kappa1*(radius-width));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "B1:=solve(eq1,B);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 103 "psi1:=-cos(Pi/2-k1*(y-radius))*Heaviside
(y-(radius-width))+B1*cosh(kappa1*y)*Heaviside(radius-width-y);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(psi1,y=0..radius); # sy
mmetric wave function" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "pl
ot(subs(y=abs(z),psi1),z=-radius..radius,title=`symmetric wave fxn`);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
83 "get wave function for antisymmetric mode:  cos (kx + c), sinh kapp
a x,  kR+c = Pi/2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "kappa:=sqrt((height-Energy)/ck);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 57 "eq1:= cos(Pi/2 - k*width) = D*sinh(kappa*
(radius-width));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "D1:=sol
ve(eq1,D);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "psi2:=-sin(k*
(y-radius))*Heaviside(y-(radius-width))+D1*sinh(kappa*y)*Heaviside(rad
ius-width-y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot(psi2,
y=0..radius); # antisymmetric wave function" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 57 "psiasm:=(Heaviside(z)-Heaviside(-z))*subs(y=abs(z)
,psi2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot(psiasm,z=-r
adius..radius,title=`anti-symm. wave fxn`);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 48 "Add symmetric and antisymm functions and animate" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "psitot:=psiasm+subs(y=abs(z),psi1);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(psitot,z=-radius..
radius);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "f1:=Energy1*1.602E-19/6.626E-34;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "f:=Energy*1.602E-19/6.626E-34;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "psitime:=psiasm*cos(f*t)+subs(y=abs
(z),psi1)*cos(f1*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plo
t(subs(t=0,psitime),z=-radius..radius);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "plot(subs(t=5E-8,psitime^2),z=-radius..radius);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "animate(psitime^2,z=-radius.
.radius,t=0..1.466666667E-7,frames=21,numpoints=150,color=blue);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "well:=Heaviside(1-abs(z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "plot(well,z=-radius..radius);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 99 "animate(\{psitime^2,well\},z=-radius..radius,t=0..1.4
66666667E-7,frames=21,numpoints=150,color=blue);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
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