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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 207 21 "Matrix Representation" }
}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 89 "We wor
k on a problem in one spatial dimension, such as the quartic double-we
ll potential." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 58 "restart; Digits:=14: with(LinearAlgebra): with(
orthopoly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7(I\"GG6\"I\"HGF$I\"LGF$I
\"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%!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sol:=solve(Vp,x);" }}
{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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\"#!)!\"\"%(DEFAULTG-%+AXESLABELSG6'-I#miG6#/I+modulenameG6\"I,Typeset
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" }}{TEXT 218 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:=20; # choosing a matrix size" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "NI_tab
le:=[seq(NI(j),j=0..N)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "77$\"/=U\"f
\"[tf!#9$\"/seS$*)QA%F%$\"/PHqY%>6#F%$\"/n,7!y>i)!#:$\"/#p(f&H$[IF,$\"
/%)onWmR'*!#;$\"/)ewVJFy#F1$\"/XP*firV(!#<$\"/O%)\\1Hf=F6$\"/Q+*3!R#Q%
!#=$\"/O$H9A$*z*!#>$\"/Gn+DA*3#F>$\"/<\\rvgkU!#?$\"/\">![Nej$)!#@$\"/z
4C(o0e\"FF$\"/_ez/r&)G!#A$\"/%R;ij75&!#B$\"/6]N\\f[()!#C$\"/p\"f:*4e9F
Q$\"/RAs\"\\`O#!#D$\"/5\"HUX*RP!#E" }}}{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 "7
7$\"#?!\"#$\"#gF%$\"$+\"F%$\"$S\"F%$\"$!=F%$\"$?#F%$\"$g#F%$\"$+$F%$\"
$S$F%$\"$!QF%$\"$?%F%$\"$g%F%$\"$+&F%$\"$S&F%$\"$!eF%$\"$?'F%$\"$g'F%$
\"$+(F%$\"$S(F%$\"$!yF%$\"$?)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 84 "# the Hamiltonian matrix can be obtained by making use of the \+
H0 eigenvalue problem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "H
M:=Matrix(N+1,N+1,shape=symmetric,datatype=float[8]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "for i from 0 to N do: for j from 0
 to i do: if i=j then HM[i+1,j+1]:=En_table[i+1]: else HM[i+1,j+1]:=0:
 fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 " HM[i+1,j+1]:= HM[i+1,j+1]
+NI_table[i+1]*NI_table[j+1]*int(phi(i)*(V-0.5*w^2*x^2)*phi(j),x=-infi
nity..infinity);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "od: od:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "En:=convert(Eigenvalues(HM),
list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "77$\"38_H\\(oL/')*!#=$\"3I!pC&
z4ns)*F%$\"3^\"=$\\$fFI!H!#<$\"3jJ.c2(f+$HF*$\"3&\\73_PImx%F*$\"3ee()e
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3\\1&RBQwpj\"F=$\"3J$>7w[5`w#F=$\"3UW)*o*zsYV$F=$\"3dL5eF^m5fF=$\"3k'4
%>zG7HqF=$\"3iWA.wSTW7!#:$\"3Yc#o+9o_U\"FN" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 62 "# some of the low-lying eigenvalue estimates come i
n 'pairs' ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Evec:=Eigenv
ectors(HM);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6
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1 0 11 "Evec[1][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"3LLG\\(oL/')*!
#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "CM:=Evec[2];" }}
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