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ll potential." }}{PARA 0 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 58 "restart; Digits:=14: with(LinearAlgebra): with(
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\"PGF$I\"TGF$I\"UGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "V:=
a*x^4-b*x^2+c;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I\"aG6\"\"\"\")I\"
xGF%\"\"%F&F&*&I\"bGF%F&)F(\"\"#F&!\"\"I\"cGF%F&" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "a:=1/32; b:=1;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "#\"\"\"\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Vp:=diff(V,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",&*&#\"\"\"\"\")F%)I\"xG6\"\"\"$F%F%*&\"\"#F%F(F%!\"\"" }
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{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!\"\"%!\"%" }}}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 11 "x0:=sol[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"
\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "c:=solve(eval(V,x=x0
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" }}{TEXT 238 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "# Defi
ne a harmonic oscillator basis:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "w:=0.4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"\"%!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "alpha:=sqrt(w);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "$\"/oL?`bCj!#9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "phi:=n->exp(-0.5*(alpha*x)^2)*H(n,alpha*x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%6$I)operatorGF%I&arrowGF%F%*&-I$
expGF%6#,$*&$\"\"&!\"\"\"\"\"*&)I&alphaGF%\"\"#F2)I\"xGF%F6F2F2F1F2-I
\"HGF%6$F$*&F5F2F8F2F2F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 87 "NI:=n->1/sqrt(int(phi(n)^2,x=-infinity..infinity)); # we could g
et this from a formula!" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F
%6$I)operatorGF%I&arrowGF%F%*$-I%sqrtG6$%*protectedGI(_syslibGF%6#-I$i
ntGF%6$*$)-I$phiGF%F#\"\"#\"\"\"/I\"xGF%;,$I)infinityGF-!\"\"F=F>F%F%F
%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "NI(1);" }}{PARA 11 "" 1
 "" {XPPMATH 20 "$\"/seS$*)QA%!#9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "N:=40; # choosing a matrix size" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "NI_tab
le:=[seq(NI(j),j=0..N)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
#H(1,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "En_table:=[seq(
(n+0.5)*w,n=0..N)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "7K$\"#?!\"#$\"#gF
%$\"$+\"F%$\"$S\"F%$\"$!=F%$\"$?#F%$\"$g#F%$\"$+$F%$\"$S$F%$\"$!QF%$\"
$?%F%$\"$g%F%$\"$+&F%$\"$S&F%$\"$!eF%$\"$?'F%$\"$g'F%$\"$+(F%$\"$S(F%$
\"$!yF%$\"$?)F%$\"$g)F%$\"$+*F%$\"$S*F%$\"$!)*F%$\"%?5F%$\"%g5F%$\"%+6
F%$\"%S6F%$\"%!=\"F%$\"%?7F%$\"%g7F%$\"%+8F%$\"%S8F%$\"%!Q\"F%$\"%?9F%
$\"%g9F%$\"%+:F%$\"%S:F%$\"%!e\"F%$\"%?;F%" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 84 "# the Hamiltonian matrix can be obtained by making \+
use of the H0 eigenvalue problem:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "HM:=Matrix(N+1,N+1,shape=symmetric,datatype=float[8])
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "for i from 0 to N do:
 for j from 0 to i do: if i=j then HM[i+1,j+1]:=En_table[i+1]: else HM
[i+1,j+1]:=0: fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 " HM[i+1,j+1]:
= HM[i+1,j+1]+NI_table[i+1]*NI_table[j+1]*int(phi(i)*(V-0.5*w^2*x^2)*p
hi(j),x=-infinity..infinity);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "od:
 od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "En:=convert(Eigenva
lues(HM),list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7K$\"3y\"4L>.1y$)*!#=
$\"3y/GU$ycy$)*F%$\"352drexN\")G!#<$\"3UQqL%)[Q\")GF*$\"3SEwa\"*GJeYF*
$\"3IqZ,8i]eYF*$\"3-NK,mi;viF*$\"3_g%)o`tr(G'F*$\"3U\"oGgoTBe(F*$\"3'3
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Rs>*3(*)*3\"F?$\"3/bz-Nu.#=\"F?$\"3%\\%47?*\\-I\"F?$\"3k!*=YxcS*R\"F?$
\"3eVmB3i*>`\"F?$\"3aVvC)[B)*o\"F?$\"3P*G[.vDD#=F?$\"3-JRS\"[.'f>F?$\"
3=**\\<b#[KR#F?$\"3gr[j#z0Wz#F?$\"39G_&)[aO)o$F?$\"3Q^s4-#zuG%F?$\"3)4
8]vQ!ejcF?$\"3moU>5'ea['F?$\"3?4%\\M^L\\\\)F?$\"3Iib'fzjDe*F?$\"3qzWeZ
\\tW7!#:$\"3QT./J-N&Q\"F\\o$\"3i5a\"GLC&*y\"F\\o$\"3)=C,%*\\\"[o>F\\o$
\"3j,%pov&>RDF\\o$\"37?Z([\"))ekFF\\o$\"3;&4>$)f>De$F\\o$\"3M.+sk?5lQF
\\o$\"3kNjZ]wd$3&F\\o$\"3&zSS0Sk*QaF\\o$\"3W1$y>\\RdW(F\\o$\"3+KsI$p+=
!zF\\o" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# some of the low
-lying eigenvalue estimates come in 'pairs' ?" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "Evec:=Eigenvectors(HM);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I
#miGF$6%Q%EvecF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q
#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*sy
mmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%'lspaceGQ,0.27
77778emF'/%'rspaceGFL-F#6%-I*mverbatimGF$6#Q^o-&I'VectorG6$%*protected
GI(_syslibG6\"6#I'columnG6\"6#/I$%idG6\"\")W64wF'-F66-Q\",F'F9F;/F?F1F
@FBFDFFFH/FKQ&0.0emF'/FNQ,0.3333333emF'-FR6#Q?-I'MatrixG6\"6#/I$%idG6
\"\")![uj(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "whattype(Ev
ec);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I(exprseqG6\"" }}}{EXCHG {PARA 0
 "> " 0 "" {MPLTEXT 1 0 11 "Evec[1][1];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "$\"3qPE$>.1y$)*!#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w
hattype(Evec[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "&I'VectorG6$%*prote
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}{TEXT 238 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "# very o
dd behaviour: for small N the lowest energy is associated with an anti
-symmetric wavefunction!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "# which eigenvalues fr
om the matrix solution can we trust? Let's graph a bunch and do an ove
rlay with the potential:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
149 "plot([V,seq(En[j],j=1..min(8,N+1))],x=-10..10,view=[-10..10,0..10
],numpoints=1000,color=[black,red,blue,green,brown,magenta,aquamarine,
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{TEXT 238 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "# to graph
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n energy." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# maybe
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