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" {TEXT -1 116 "In order to test the linear Stark effect in the degene
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lF_ilF(-F$6$7_oF'7$F<$\"3)RQy^vitT$F\\[n7$FF$\"3rHh*y/bJk%FU7$FL$\"34`
J%*)fXC]*FU7$FQ$\"31R9\\N.13<Fdz7$FW$\"3Wt]Ylqj(z#Fdz7$Ffn$\"3KBi6A9&G
F%Fdz7$F`o$\"3#oY$pB8*[A*Fdz7$Fjo$\"3]'*eY*z40n\"F07$Fhq$\"3#eH4Z'y?\"
p#F07$Fbr$\"3?IN6Ra%3(RF07$F\\s$\"3U#**4vv\\RZ&F07$Ffs$\"3%*oG&*)[Qb;(
F07$F[t$\"3ms$[i[^)f))F07$F`t$\"3Mnhu\"[PK1\"F-7$Fet$\"3Y_>^8X\\]7F-7$
Fjt$\"3AHQG[d1P9F-7$F_u$\"3Er?V$>l[i\"F-7$Fdu$\"34UPIg^S.=F-7$Fiv$\"3#
zYL:#\\uo>F-7$F]x$\"3g/&*HZQu=@F-7$Fbx$\"3;(*4<)pk[D#F-7$Fgx$\"3;1W_'[
T5P#F-7$Fay$\"3#GCUxap[_#F-7$Ffy$\"3w#p1-R'y#e#F-7$F[z$\"3j!R![#eN1i#F
-7$$\"30++]-g0L:F[v$\"3)fn?zrUBj#F-7$F`z$\"3CG9$*piNREF-7$$\"3%)****\\
<w-)f\"F[v$\"35AaV![L=k#F-7$Ffz$\"3(ft.fhW*REF-7$F[[l$\"3du*[d!RmCEF-7
$F`[l$\"3gjm<\"pE_f#F-7$Fe[l$\"3ov+WAOXdDF-7$Fj[l$\"3c)*QB+dK5DF-7$Fd
\\l$\"3!=Kx.=*=nBF-7$F^]l$\"3N%[Qx/9-A#F-7$Fc]l$\"3d(o-QTN2.#F-7$F]^l$
\"3G\\ho>0ja=F-7$Fa_l$\"3y/IS)*)\\*f;F-7$F[`l$\"3C!yK&p^uy9F-7$Fe`l$\"
3c1([Jcb%)H\"F-7$Fj`l$\"3<1(QgffM9\"F-7$F_al$\"3ibs3WR?%*)*F07$Fdal$\"
3#oFJBMW'[%)F07$Fial$\"3RBOrECW?tF07$F^bl$\"3eUJyxs?NiF07$Fcbl$\"3]7\"
)zt&pDD&F07$Fhbl$\"3QD_itT6=WF07$F]cl$\"3D/D:U`.?PF07$Fbcl$\"3k-B%49b#
eIF07$Fgcl$\"3=-,UT01aDF07$F\\dl$\"3ew@K(Q%f)4#F07$Fadl$\"3ol2>\"*)f/v
\"F07$Ffdl$\"3\\6=G(y\"eI9F07$F[el$\"3%y3#4)3j%z6F07$F`el$\"3TFi[Q&\\*
4'*Fdz7$Feel$\"3()e@(\\!o9VyFdz7$Fjel$\"3L0%)3Zx7AjFdz7$F_fl$\"3;*Gw!)
zDJ7&Fdz7$Fdfl$\"3gMbQMIA@TFdz7$Fifl$\"3=D$e%=<;8LFdz7$F^gl$\"3,Jn-Ahe
0FFdz7$Fcgl$\"3b&[FSY\"3S@Fdz7$Fhgl$\"3s+uznNxJ<Fdz7$F]hl$\"3ujgxxT9z8
Fdz7$Fbhl$\"3F\\(Gu#=226Fdz7$Fghl$\"3`_7**fVaD()FU-F\\il6&F^ilF(F(F_il
-%+AXESLABELSG6$Q\"r6\"Q!Fcho-%&TITLEG6#Q>Hydrogen(n=4)~radial~orbital
sFcho-%%VIEWG6$;F(Fghl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "These radial functions are normali
zed. Are they orthogonal?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "int(expand(R(2,0,r)*R(1,0,r)*r^2),r=0..infinity) assuming Z>0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$,$*&#\"\"\"\"\"$F)**)%\"ZGF
*F)-%$expG6#,$**F*F)\"\"#!\"\"F-F)%\"rGF)F4F)F3#F)F3)F5F*F)F)F)/F5%)in
finityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Maple should not have
 trouble with this! For some reason maple9.5 doesn't figure this one o
ut. The problem is with the symbol Z." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "int(eval(expand(R(2,0,r)*R(1,0,r)),Z=1)*r^2,r=0..infi
nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 83 "The angular parts of the wavefunctions are spherical
 harmonics. Let us define them:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "with(orthopoly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%\"GG%\"
HG%\"LG%\"PG%\"TG%\"UG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "P
lm:=proc(theta,l::nonnegint,m::integer) local x,y,f;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "x:=cos(theta);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "if m>0 then f:=subs(y=x,diff(P(l,y),y$m));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "else f:=subs(y=x,P(l,y)); fi;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "(-1)^m*sin(theta)^m*f; end:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "Plm(theta,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,$*(\"#:\"\"\")-%$sinG6#%&thetaG\"\"#F&-%$cosGF*F&F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 75 "For the spherical harmonics we don't need
 the Plm's with negative argument." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "Y:=proc(theta,phi,l::nonnegint,m::integer) local m1;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "m1:=abs(m); if m1>l then RETURN
(\"|m\} has to be <= l for Y_lm\"); fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 78 "exp(I*m*phi)*Plm(theta,l,m1)*(-1)^m*sqrt((2*l+1)*(l-m1)!/(4*Pi
*(l+m1)!)); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta
,phi,3,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*(,&*&
#\"\"&F'F&*$)-%$cosG6#%&thetaG\"\"$F&F&F&*&#F3F'F&F/F&!\"\"F&\"\"(F%%#
PiG#F6F'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Y(theta,phi
,3,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"#7F&*,-%$expG6#
*&%$phiGF&^#F&F&F&-%$sinG6#%&thetaGF&,&*&#\"#:\"\"#F&*$)-%$cosGF1F7F&F
&F&#\"\"$F7!\"\"F&\"#@#F&F7%#PiG#F>F7F&F&" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 108 "No:=(l,m)->int(int(evalc(Y(theta,phi,l,m)*conjugat
e(Y(theta,phi,l,m))),phi=0..2*Pi)*sin(theta),theta=0..Pi);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#NoGj+6$%\"lG%\"mG6\"6$%)operatorG%&arrowG
F)-%$intG6$*&-F.6$-%&evalcG6#*&-%\"YG6&%&thetaG%$phiG9$9%\"\"\"-%*conj
ugateG6#F7F>/F;;\"\"!,$*&\"\"#F>%#PiGF>F>F>-%$sinG6#F:F>/F:;FDFHF)F)F)
6$FDFD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "No(1,0);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "No(1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Define an inner product for the a
ngular degrees of freedom:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "IP:=(f,g)->int(int(evalc(g*conjugate(f)),phi=0..2*Pi)*sin(theta),t
heta=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IPGj+6$%\"fG%\"gG6
\"6$%)operatorG%&arrowGF)-%$intG6$*&-F.6$-%&evalcG6#*&9%\"\"\"-%*conju
gateG6#9$F8/%$phiG;\"\"!,$*&\"\"#F8%#PiGF8F8F8-%$sinG6#%&thetaGF8/FH;F
@FDF)F)F)6$F@F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "IP(Y(the
ta,phi,1,0),Y(theta,phi,3,0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "The required matrix element \+
involves as the operator the displacement" }{TEXT 19 2 " z" }{TEXT -1 
14 " expressed as " }{TEXT 19 13 " r*cos(theta)" }}{PARA 0 "" 0 "" 
{TEXT -1 82 "The radial part of the matrix element is defined first. W
e need to set the charge:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 
"Z:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZG\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "RME:=(n,l,np,lp)->int(expand(r^3*R(
n,l,r)*R(np,lp,r)),r=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$RMEGj+6&%\"nG%\"lG%#npG%#lpG6\"6$%)operatorG%&arrowGF+-%$intG6$-%'e
xpandG6#*()%\"rG\"\"$\"\"\"-%\"RG6%9$9%F7F9-F;6%9&9'F7F9/F7;\"\"!%)inf
inityGF+F+F+6&FEFEFEFE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "e
valf([RME(2,1,1,0),RME(3,1,1,0),RME(4,1,1,0),RME(5,1,1,0)]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&$\"/*f>?m-H\"!#8$\"/L=EC*o;&!#9$\"/Z#*Q!Qe
/$F)$\"/=Nh\"Rq3#F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Now the pa
rt where the polar and azimuthal angles " }{TEXT 19 5 "theta" }{TEXT 
-1 5 " and " }{TEXT 19 3 "phi" }{TEXT -1 16 " are integrated:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "AME:=(l,m,lp,mp)->IP(Y(theta
,phi,l,m),cos(theta)*Y(theta,phi,lp,mp));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$AMEGj+6&%\"lG%\"mG%#lpG%#mpG6\"6$%)operatorG%&arrowG
F+-%#IPG6$-%\"YG6&%&thetaG%$phiG9$9%*&-%$cosG6#F5\"\"\"-F36&F5F69&9'F=
F+F+F+6&\"\"!FCFCFC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "AME(
1,0,0,0),AME(2,0,0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&\"\"$!\"
\"F%#\"\"\"\"\"#F(\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 11 "Exerc
ise 1:" }}{PARA 0 "" 0 "" {TEXT -1 26 "Test the selection rules: " }
{TEXT 19 11 "Delta-m = 0" }{TEXT -1 6 "  and " }{TEXT 19 18 "Delta-l =
 0, \\pm 1" }{TEXT -1 35 " by trying various combinations of " }{TEXT 
19 5 "(l,m)" }{TEXT -1 6 " with " }{TEXT 19 7 "(l',m')" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 289 "Now \+
we assemble a Hamiltonian matrix using the hydrogenic bound states. Th
ere is a danger in this approach: the bound hydrogen states are not a \+
complete set. Results will converge, but not necessarily towards the c
orrect eigenvalue. This problem shall be left to a mini-project at the
 end." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 
"We use the fact that the problem retains axial symmetry (rotations ab
out the z-axis), and pick a value for " }{TEXT 19 1 "m" }{TEXT -1 1 ":
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=0;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"mG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "
There are two indeces that control the coverage of Hilbert space: a cu
t-off in the principal quantum number, and one in the angular momentum
 values to be used:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "n_max
:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&n_maxG\"\"&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "l_max:=4;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&l_maxG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "
How many states are there in this sector? We can make a look-up table:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "i_count:=0:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 25 "for n from 1 to n_max do:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "for l from 0 to n-1 do:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "i_count:=i_count+1: LU[i_count]:=[n,l]: RLU[n,l]:=i_c
ount:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "od: od: N:=i_count;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 73 "The H0 part of the Hamiltonian is given by the following \+
diagonal matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Hmat:=M
atrix(N,N,shape=symmetric,datatype=float[8]):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 59 "for i from 1 to N do: n:=LU[i][1]: Hmat[i,i]:=-0
.5/n^2; od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "For the added inte
raction we pre-calculate the matrix without the field strength paramet
er:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Wmat:=Matrix(N,N,sha
pe=symmetric,datatype=float[8]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
153 "for i from 1 to N do: n:=LU[i][1]: l:=LU[i][2]: for j from 1 to i
 do: np:=LU[j][1]: lp:=LU[j][2]: Wmat[i,j]:=evalf(RME(n,l,np,lp)*AME(l
,m,lp,m)); od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "HmatF
:=Matrix(Hmat+0.001*Wmat,shape=symmetric);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&HmatFG-%'RTABLEG6+\")))o\"H#&%&floatG6#\"\")%'Matrix
G&%+triangularG6#%&upperG%.Fortran_orderG7#%*symmetricG\"\"#;\"\"\"\"#
:F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "convert(Eigenvalues(
HmatF),list);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#71$!3UcJfjx,+]!#=$!3=
!QRef&z!G\"F&$!3zAM!HbX2A\"F&$!3AT,YiYE`l!#>$!3$[aFyYfGk&F-$!3cY%o]xU/
f&F-$!3<LooR.B=ZF-$!3k#)ea'QRDL%F-$!3GL<%RA/#eTF-$!3+9sJk\"ef/$F-$!3Wn
Aa=7&yz#F-$!3:e'3[*R?+<F-$!3'f>6<Nv/[\"F-$!3W)zZ6#R(y'H!#?$\"3E7vrL%Rd
;\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "We now assemble a table
 of eigenvalues for a graphical presentation of the energy levels as a
 function of field strength " }{TEXT 19 1 "F" }{TEXT -1 84 ". We appli
ed trial and error to see what would be a reasonable field strength ra
nge." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Np:=200:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for nu from 1 to Np do:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 57 "F:=0.0001*nu; HmatF:=Matrix(Hmat+F*Wmat,sha
pe=symmetric);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "FV[nu]:=F; EVal[n
u]:=convert(Eigenvalues(HmatF),list); od:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 168 "plot([seq(seq([FV[nu],EVal[nu][i]],i=1..N),nu=1..N
p)],style=point,view=[0..0.0001*Np,-0.55..0],title=\"Energy levels as \+
a function of field strength F in atomic units\");" }}{PARA 13 "" 1 "
" {GLPLOT2D 1008 501 501 {PLOTDATA 2 "6'-%'CURVESG6$7dfv7$$\"31+++++++
5!#@$!35Bcix,++]!#=7$F($!3g0;G.x+`7F-7$F($!3kT\"4Nl2qC\"F-7$F($!3kDbx?
fJYc!#>7$F($!3)*=AI+9QcbF77$F($!3'HDudn%GmaF77$F($!3Z(3ZL]h*3LF77$F($!
3I7/,D\"f#*=$F77$F($!3#QDn'z9&*oIF77$F($!3cMX2D&*=[HF77$F($!3!\\>DY!G
\"fH#F77$F($!3aP_&>x)HY@F77$F($!3SADJBw`'*>F77$F($!3K\"**RL7?n%=F77$F(
$!3)*H)=$p$)p'p\"F77$$\"35+++++++?F*$!3]_P]52++]F-7$FZ$!3iGE*f\"4.c7F-
7$FZ$!3/M&[x^ISC\"F-7$FZ$!3I4*fPNl'QdF77$FZ$!3?>r)ocj)ebF77$FZ$!3eC:VT
[Ty`F77$FZ$!3\\MCF()*RF]$F77$FZ$!3q)=@DReEE$F77$FZ$!3uar')*GG--$F77$FZ
$!3?1AemUawFF77$FZ$!3/n.XDgr\"e#F77$FZ$!318%[(y,K%G#F77$FZ$!3S[$[q8;i)
>F77$FZ$!3+b$)Ha1l(o\"F77$FZ$!3q:1NIv-)Q\"F77$$\"3u**************HF*$!
3VF\"Q')f,++&F-7$Fcq$!3=&)*=vzp!f7F-7$Fcq$!3=vC3`%o5C\"F-7$Fcq$!3i>'3y
B;F$eF77$Fcq$!3+BCnfh,jbF77$Fcq$!3cBn\")ev'=H&F77$Fcq$!3jH3A2\\c4PF77$
Fcq$!31Udw#)H1YLF77$Fcq$!3!z#>6fB/yHF77$Fcq$!3WVnw7Y2aGF77$Fcq$!3^6gsg
@l3EF77$Fcq$!3:LY7(40GT#F77$Fcq$!3y-v`luAp>F77$Fcq$!3#G&p6nJ\"Q_\"F77$
Fcq$!3%4!GglaUv5F77$$\"3?+++++++SF*$!37&*\\.UG++]F-7$Fbt$!3%p**zT^C@E
\"F-7$Fbt$!3tz7;E87Q7F-7$Fbt$!3@7+T&='fGfF77$Fbt$!3?rYN3M')obF77$Fbt$!
3:1Fz!Qtl?&F77$Fbt$!3MA$G^q3E$RF77$Fbt$!3)\\)p`HJ9SMF77$Fbt$!3Y6#*Rfcd
4JF77$Fbt$!3utA!33N<%HF77$Fbt$!3`*)oP%*)H2`#F77$Fbt$!3#=%ea#)y_VCF77$F
bt$!3Gh\\8pQ(e%>F77$Fbt$!3K'Rmun[gN\"F77$Fbt$!37bzb*pR!*f(!#?7$$\"37++
+++++]F*$!38%4.2W/++&F-7$Fbw$!3=?LVR_>l7F-7$Fbw$!3\\6iz4!*=N7F-7$Fbw$!
3(zle^\"=XEgF77$Fbw$!3WG=Zp3WwbF77$Fbw$!3BdE%H@rC7&F77$Fbw$!3kD/A%>@Q<
%F77$Fbw$!3wox#yJ`aa$F77$Fbw$!3;HV!e2!fXLF77$Fbw$!3TpZOAv(3\"HF77$Fbw$
!3\"=eS]JGuj#F77$Fbw$!3U];5^d\\!G#F77$Fbw$!38]50qq`;>F77$Fbw$!3S\\brI[
.&=\"F77$Fbw$!35wGTEzx@WF`w7$$\"3[**************fF*$!3#*pOl%R1++&F-7$F
az$!3Av25a@Go7F-7$Faz$!3oiu%GQrAB\"F-7$Faz$!3jSEV1]XEhF77$Faz$!3cYE&)z
tz&e&F77$Faz$!3PN9C!y2&R]F77$Faz$!3@&y&RnI#HV%F77$Faz$!3Q7NKd%oQm$F77$
Faz$!3))>C#*>fsfNF77$Faz$!3=L]L<O%e)GF77$Faz$!3O&)\\'pyF@t#F77$Faz$!3L
MJ\\0a8>@F77$Faz$!3k0#*y9Ch\")=F77$Faz$!3(=\"pl%o78,\"F77$Faz$!3]-.S-$
yvA\"F`w7$$\"3%***************pF*$!3fi/!Rq3++&F-7$F`]l$!3[&o<mW&Qr7F-7
$F`]l$!3@&pC,Lo$H7F-7$F`]l$!3HT0GJR\")GiF77$F`]l$!3elH%R:)*pf&F77$F`]l
$!3gNBcvsjd\\F77$F`]l$!3xRnc$Q>vq%F77$F`]l$!39:kZ+M'o!QF77$F`]l$!34Yn,
A?]TPF77$F`]l$!3eF$H\"ehptGF77$F`]l$!3)>)*R'yk)z!GF77$F`]l$!3yGv)p\"eZ
f>F77$F`]l$!3;j-zx1AT=F77$F`]l$!3#)fm'R$p8`$)F`w7$F`]l$\"3o[:^:8&)z>F`
w7$$\"3S+++++++!)F*$!3\\A(f%o8,+]F-7$F_`l$!3Md*3LJ0XF\"F-7$F_`l$!3R,)*
HU(zkA\"F-7$F_`l$!3&[E$*z\"QzLjF77$F_`l$!3\\V6YS*H,h&F77$F_`l$!3!HI&[;
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G6$Q!6\"Fhg_m-%&STYLEG6#%&POINTG-%&TITLEG6#QjnEnergy~levels~as~a~funct
ion~of~field~strength~F~in~atomic~unitsFig_m-%%VIEWG6$;Fcg_m$\"$+#!\"%
;$!#b!\"#Fcg_m" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "We observe at the \+
level of " }{TEXT 19 3 "n=2" }{TEXT -1 84 " a linear splitting of the \+
eigenvalues for the two correct zeroth order eigenstates " }{TEXT 19 
10 "|2s>-|2p0>" }{TEXT -1 5 " and " }{TEXT 19 10 "|2s>+|2p0>" }{TEXT 
-1 71 " respectively as derived in first-order degenerate perturbation
 theory." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
102 "The diagram illustrates the justification of first-order perturba
tion theory for weak fields: the two " }{TEXT 19 7 "n=2,m=0" }{TEXT 
-1 244 " levels at first evolve with a large energy separation from th
e other levels. Energy denominators in second-order corrections suppre
ss couplings. It is justified to ignore them in this regime and calcul
ate the linear Stark effect in PT. Beyond " }{TEXT 19 7 "F=0.002" }
{TEXT -1 183 " atomic units this begins to be questionable, as an uppe
r level is getting close. Eventually, with increasing field strength d
eviations from the linear behavior become quite apparent." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We can also notic
e the much stronger effect that the electric field has on the higher-l
ying states. At the " }{TEXT 19 3 "n=3" }{TEXT -1 49 " level there are
 three states, because there are " }{TEXT 19 18 "|3s>, |3p0>, |3d0>" }
{TEXT -1 14 " to be mixed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 52 "The upward moving energy levels cross over into
 the " }{TEXT 19 3 "E>0" }{TEXT -1 34 " regime at some coupling consta
nt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 225 "L
evel crossings (avoided crossings?) occur at strong fields. True level
 crossings involve a degeneracy which would be indicative of a symmetr
y in the Hamiltonian. Therefore, when a Hamiltonian depends on a param
eter (such as " }{TEXT 19 1 "F" }{TEXT -1 816 " in our case) true leve
l crossings should not occur when the parameter is varied. Levels can \+
get very close but do repel each other to not become truly degenerate.
 Avoided level crossings play an important role in molecular phenomena
 and in slow ion-atom collisions. The character of a wavefunction (e.g
., dipole moment) is usually maintained across the avoided crossing. F
or a system to follow its level as the parameter is changed, involves \+
changing character. This will not happen when the parameter is varied \+
rapidly: in this case the evolution follows by maintaining wavefunctio
n shape and crossing the level. Diabatic energy level diagrams have al
lowed crossings, they do not represent solutions of the Schroedinger e
quation with parametric dependence, but reflect how the system evolves
 in real circumstances." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 120 "The ground state is not much affected for the field
 strength shown, because it sits deeply inside the potential barrier.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We l
earned from the treatment in parabolic coordinates that the calculated
 real energies are locations of resonances." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 11 "Exercise 2:" }}{PARA 0 "" 0 "
" {TEXT -1 129 "Select a field strength regime with an apparent energy
 level crossing which is isolated from the other neighbors (for exampl
e at " }{TEXT 19 7 "F=0.003" }{TEXT -1 5 " the " }{TEXT 19 3 "n=2" }
{TEXT -1 113 " upper level meets one of the levels from above). Invest
igate this crossing by running the calculation on a fine " }{TEXT 19 
1 "F" }{TEXT -1 169 "-grid, and by zooming in on this region in the gr
aph. Do you find that the levels get close, but do not cross as is vis
ible for some of the higher-level curve crossings?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 13 "Mini-Project:" }}{PARA 
0 "" 0 "" {TEXT -1 152 "Repeat the calculation with a Laguerre-type ba
sis instead of the bound hydrogenic states. In this type of basis the \+
exponential factor is fixed for all " }{TEXT 19 1 "n" }{TEXT -1 210 "-
levels. Compare the convergence of a few low-lying eigenvalues at fixe
d field strength in this basis with the converged calculations in the \+
hydrogenic bound-state basis. Describe and explain your observations.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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