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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart; Digits:=14
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# Schroedinger QM in 1
d: a double-well potential." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "# hbar=m=1 units + one more freedom" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "V:=a*x^4-b*x^2+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(
*&I\"aG6\"\"\"\")I\"xGF%\"\"%F&F&*&I\"bGF%F&)F(\"\"#F&!\"\"I\"cGF%F&" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "# choose c such that pote
ntial is a minimum at E=0 (classical reference point)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a:=1/16; b:=1/2;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "#\"\"\"\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"\"\"\"
\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Vp:=diff(V,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 ",&*&#\"\"\"\"\"%F%)I\"xG6\"\"\"$F%F%F(!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sol0:=solve(Vp,x);"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!\"\"#!\"#" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "x0:=sol0[2];" }}{PARA 11 "" 1 "" {XPPMATH 20
 "\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "c:=solve(eval(V,
x=x0),c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 46 "plot(V,x=-2*x0..2*x0,view=[-2*x0..2*x0,0..2]
);" }}{PARA 13 "" 1 "" {TEXT 216 0 "" }{GLPLOT2D 824 400 400 
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{MPLTEXT 1 0 67 "# Question: will the ground state energy be well belo
w the barrier?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Vb:=eval(
V,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> 
" 0 "" {MPLTEXT 1 0 63 "# what controls this question?  [Heisenberg, z
ero-point energy]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# what
 could give us a clue?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "V
t:=taylor(V,x=x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "+),&I\"xG6\"\"\"\"
\"\"#!\"\"F&F'#F&F'\"\"$#F&\"#;\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 140 "# if we are interested in a pair of deep wells, then
 the Harmonic Oscillator approximation will tell us what the zero poin
t energy might be." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "# he
re omega=sqrt(2) [since V=0.5*w^2*x^2 is the standard form], and we do
 not expect a zero-point energy below Vb  0.5*omega vs Vb!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "taylor(V,x=x0,3);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "+',&I\"xG6\"\"\"\"\"\"#!\"\"F&F'-I\"OG%*protectedG6#F&
\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Vt2:=convert(taylo
r(V,x=x0,3),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$),&I\"xG6\"\"
\"\"\"\"#!\"\"F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "w:=sq
rt(solve(Vt2=0.5*w2*(x-x0)^2,w2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$
\"/JPiN@99!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "# so how d
o we find out what the energies are for the ground state, and the firs
t excited state?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "# pick \+
a trial energy in accord with the above estimate:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "Et:=0.5*w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"/b'=\"y1rq!#9" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "SE:=-diff(psi(x),x$2)+(V-Et)
*psi(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "/,&-I%diffG%*protectedG6$-
F%6$-I$psiG6\"6#I\"xGF,F.F.!\"\"*&,(*&#\"\"\"\"#;F4)F.\"\"%F4F4*&#F4\"
\"#F4)F.F:F4F/$\"/X8)=K*GH!#9F4F4F*F4F4\"\"!" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 26 "ICe:=psi(0)=1,D(psi)(0)=0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$/-I$psiG6\"6#\"\"!\"\"\"/--I\"DG6$%*protectedGI(_syslib
GF&6#F%F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sol:=dsolve(
\{SE,ICe\},numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 23 "psi0:=eval(psi(x),sol):" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 10 "psi0(0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"3jg^9
F9AM5!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "# seems to work
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot(psi0(x),x=-3*x0..3
*x0,view=[-2.5*x0..2.5*x0,-2..2],numpoints=1000);" }}{PARA 13 "" 1 "" 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Et:=0.5;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"\"&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 
"SE:=-diff(psi(x),x$2)+(V-Et)*psi(x)=0;" }}{PARA 11 "" 1 "" {XPPMATH 
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{MPLTEXT 1 0 15 "# seems to work" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "# this is too low." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# So let us look at a way to
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{MPLTEXT 1 0 24 "# Define a large x value" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "X:=4*x0
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{MPLTEXT 1 0 142 "# Define a function that takes Et as an input value,
 sets up the Schroedinger solution, and returns the value of the magni
tude squared at x=X." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "psiX:=proc(Et) local SE,ICe
,sol,psi0; global V,X;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 38 "SE:=-diff(psi(x),x$2)+(V-Et)*psi(x)=0;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ICe:=psi(0)=1,D(psi)(0)=0;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sol:=dsolve(\{SE,ICe\},numeric,outp
ut=listprocedure):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "psi0:=eval(ps
i(x),sol):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "psi0(X)^2; end:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 10 "psiX(0.6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"/2b7[
R;B\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# Now generate a
 sequence of points where the trial energy is varied." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "PP:=seq([0.55+0.01*i,psiX(0.55+0.01
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "# we have the ground state c
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