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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 45 "Time Dependent Solution
s for Motion on a Ring" }}{PARA 0 "" 0 "" {TEXT 257 35 "by Corinne Man
ogue and Kerry Browne" }}{PARA 0 "" 0 "" {TEXT -1 30 "Copyright 2008 C
orinne Manogue" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart: \+
with(plots):\nassume(phi, real, t, real):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Define the normalized spa
tial eigenfunctions (" }{XPPEDIT 18 0 "Phi[m];" "6#&%$PhiG6#%\"mG" }
{TEXT -1 38 ") for the particle confined to a ring." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 33 "Phi:=(1/sqrt(2*Pi))*exp(I*m*phi);" }{TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
105 "Multiply the spatial eigenfunctions by their time dependence to o
btain the time dependent eigenfunctions." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "PhiT:=Phi*exp(-I*m^2*t);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "Here are some sample wav
efunctions.  Each of these consists of a linear combination of time de
pendent eigenstates on the ring.  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "PsiT:=subs(m=1,PhiT);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "PsiT:=1/sqrt(2)*(subs(m=1,PhiT)+subs(m=2,PhiT));" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 63 "PsiT:=1/sqrt(3)*(subs(m=1,PhiT)+subs(m=2,PhiT)+s
ubs(m=3,PhiT));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "PsiT:=1/
sqrt(4)*(subs(m=1,PhiT)+subs(m=2,PhiT)+subs(m=-1,PhiT)+subs(m=-2,PhiT)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "PsiT:=1/sqrt(6)*(sub
s(m=1,PhiT)+subs(m=2,PhiT)+subs(m=3,PhiT)+subs(m=-1,PhiT)+subs(m=-2,Ph
iT)+subs(m=-3,PhiT));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 154 "\"Square\" your chosen wavefunction to g
et the probability density.  \nYou must execute the commands starting \+
here, each time you choose a new wavefunction. " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "PsiTsq:=simplify((abs(PsiT))^2);\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 70 "Define the plotting period and format the wavef
unction for plotting.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "
PsiTsqcolor:=subs(t=i*2*Pi/101,(PsiTsq)):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 149 "Plot an animation of th
e wavefunction on a standard cartesian plot.  In this plot the probabi
lity density is indicated by the height of the function." }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 60 "Psianimcart:=[seq(plot(PsiTsqcolor, phi=0..2*
Pi),i=1..100)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display(Psianimc
art, insequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 106 "Plot  an animation of the wavefunction u
sing color to indicate the probability density (red =0, purple =1)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "Psianimcyl:=[seq(cylinderplot([r, \+
phi, 0], \nphi=0..2*Pi,r=.98..1.02, color=PsiTsqcolor,\naxes=boxed, sc
aling=constrained, style=patchnogrid),i=1..100)]:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 77 "display(Psianimcyl, insequence=true,scaling=constra
ined, orientation=[0,0]);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Pl
ot an animation of the wavefunction using both height and color to ind
icate probability density." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
157 "Psianimboth:=[seq(cylinderplot([1,phi,z],\nphi=0..2*Pi,z=0..subs(
t=i*2*Pi/99,PsiTsq),\ncolor=PsiTsqcolor,numpoints=625,axes=boxed,style
=patchnogrid),i=1..99)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display
(Psianimboth, insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{MARK "0 2 0" 14 }{VIEWOPTS 0 1 0 3 2 1804 1 1 1 1 }
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