{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 " Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O utput" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 34 "Time-dependent perturbati on theory" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 326 "We consider a finite-time monochromatic perturbation which connec ts an initially populated ground state to higher energy levels. We sho uld keep in mind that a single-frequency sinusoidal signal when evalua ted over finite times is not monochromatic at all, but experiences a b roadening and other effects due to the pulse shape." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "For simplicity we consi der harmonic oscillator eigenstates." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart; Digits:=14: with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "w:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "hbar:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%hbarG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "En:=n->(n+1/2)*hbar*w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EnG j+6#%\"nG6\"6$%)operatorG%&arrowGF(*(,&9$\"\"\"#F/\"\"#F/F/%%hbarGF/% \"wGF/F(F(F(6#\"\"!" }}}{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 31 "An:=n->1/sqrt (2^n*n!*sqrt(Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AnGj+6#%\"nG6 \"6$%)operatorG%&arrowGF(*&\"\"\"F--%%sqrtG6#*()\"\"#9$F--%*factorialG 6#F4F--F/6#%#PiGF-!\"\"F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "beta:=sqrt(m*w/hbar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "phi :=n->An(n)*exp(-beta^2*x^2/2)*H(n,beta*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiGj+6#%\"nG6\"6$%)operatorG%&arrowGF(*(-%#AnG6#9$ \"\"\"-%$expG6#,$*(\"\"#!\"\"%%betaGF7%\"xGF7F8F1-%\"HG6$F0*&F9F1F:F1F 1F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "#plot([se q(phi(n),n=0..3)],x=-5..5,color=[red,blue,green,black],numpoints=500); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Normalization check:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(phi(3)^2,x=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 188 "Suppose the interaction represents a sinusoidally varying homogeneous electri c field. The strength parameter lambda is unspecified, and the circula r frequency of the field is called alpha." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "W:=x*sin(alpha*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WG*&%\"xG\"\"\"-%$sinG6#*&%&alphaGF'%\"tGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Wmat:=(m,n)->int(phi(m)*W*phi(n),x=-infin ity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WmatGj+6$%\"mG% \"nG6\"6$%)operatorG%&arrowGF)-%$intG6$*(-%$phiG6#9$\"\"\"%\"WGF5-F26# 9%F5/%\"xG;,$%)infinityG!\"\"F>F)F)F)6$\"\"!FA" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "Wmat(0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&#\"\"\"\"\"#F&*&-%$sinG6#*&%&alphaGF&%\"tGF&F&F'F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Wmat(0,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Wma t(0,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 119 "The interaction connects only states of opposite \+ parity. This is obvious from the symmetry properties of the integrand. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "From the calculation it also becomes clear that the interaction connects o nly neighboring levels. This is a special feature of the harmonic osci llator eigenfunctions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "Note that we have two frequencies in our problem: t he harmonic oscillator frequency w=1, and the frequency of the field u sed to excite the system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 59 "Let us make a choice for the frequency of the pert urbation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "alpha:=w*0.999 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG$\"$***!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "T_field:=2*Pi/alpha;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(T_fieldG,$*&$\"/?+-?+-?!#8\"\"\"%#PiGF*F*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "N_cyc:=3;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&N_cycG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "T:=N_cyc*T_field;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,$* &$\"/g+1g+1g!#8\"\"\"%#PiGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "lambda:=0.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG$\" \"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "We parametrized the interaction matrix element in terms of time t. 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=Fi^p$\"/(RR(Q([U%F:7$$\"/E.*fS*[=Fi^p$\"/GXZ2))HWF:7$$\"/&f%ot&H&=Fi^ p$\"/WCirMMWF:7$$\"/bz!*HVc=Fi^p$\"/#es#*puV%F:7$$\"/aSp[Ug=Fi^p$\"/o% )\\*)GSWF:7$$\"/(e^WxT'=Fi^p$\"/tp!GtAW%F:7$$\"/D7vP*y'=Fi^p$\"/hO2uoV WF:7$$\"/YXhIhr=Fi^p$\"/xk`(RYW%F:7$$\"/`3)\\1a(=Fi^p$\"/u%yNE_W%F:7$$ \"/SOxuUz=Fi^p$\"//CO;`XWF:7$$\"/Yh@V6$)=Fi^p$\"/AsrYiXWF:7$$\"/\")eMC %o)=Fi^p$\"/(o(RZjXWF:-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$ Q\"s6\"Q!Fiht-%%VIEWG6$;F($\"/&)eMC%o)=Fi^p%(DEFAULTG" 1 2 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 36 "How should we interpret this result?" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "1) We can only exc ite the neighboring " }{TEXT 19 3 "k=1" }{TEXT -1 133 " level in first -order perturbation theory. However, we should expect population of hi gher states as a result of second-order effects." }}{PARA 0 "" 0 "" {TEXT -1 204 "2) It appears as if the level will not only be excited, \+ but also de-excited by the same matrix element. At the end of a comple te cycle we find a complete return of the population back to the groun d state." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 98 "Modify the frequency o f the perturbation, i.e., choose alpha to be not as close to the reson ance (" }{TEXT 19 6 "hbar*w" }{TEXT -1 73 " is the energy difference b etween neighboring levels, i.e., also between " }{TEXT 19 3 "k=0" } {TEXT -1 5 " and " }{TEXT 19 3 "k=1" }{TEXT -1 210 "). Start with smal l deviations from resonance before you choose quite different circular frequencies for the 'laser pulse'. You will understand better what ha ppens away from resonance in some of the work below." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "The strong excitation results (on resonance) should make us suspicious about the usefulness of first-order PT in this problem. After all the zeroth-order result for " }{TEXT 19 5 "c0(t)" }{TEXT -1 86 " was used, namely a probability of 1 in the original state thro ughout the interaction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Let us attempt some second-order calculations for th e population of the levels " }{TEXT 19 7 "k=0,1,2" }{TEXT -1 132 ". Fo r this purpose it is not sufficient to record the transition probabili ties, but we need to keep track of the complex amplitudes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "To order O1 we ha ve for the only non-vanishing excited-state probability amplitude:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "c1_O1:=unapply(-I/hbar*int( exp(I*(En(1)-En(0))*t)*Wmat(0,1),t=0..s),s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&c1_O1Gj+6#%\"sG6\"6$%)operatorG%&arrowGF(*&^#!\"\"\" \"\",(*($\"/Tla_wLN!#6F/-%$expG6#*&^#$F/\"\"!F/9$F/F/-%$cosG6#,$*&$\"/ +++++!***!#9F/F " 0 "" {MPLTEXT 1 0 74 "c0_O1:= 1; # this is not very logical, as it is the result up to 1st order." } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&c0_O1G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "c2_O2:=unapply(-I/hbar*int(exp(I*(En(2)-En(1) )*t)*Wmat(1,2)*c1_O1(t),t=0..s),s);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%&c2_O2Gj+6#%\"sG6\"6$%)operatorG%&arrowGF(*&^#!\"\"\"\"\",.^#$!/(e. s$**H))!\"*F/*(^#$!/>vwp$)Q))F4F/-%$expG6#*&^#$\"\"#\"\"!F/9$F/F/-%$co sG6#,$*&$\"/+++++)*>!#8F/FAF/F/F/F/*($\"/=\"HbK)Q))F4F/F9F/-%$sinGFDF/ F.*&^#$\"/&GIRcK%))!#7F/F9F/F/*(^#$\"/=2W()*fw\"!\")F/-F:6#*&^#$F/F@F/ FAF/F/-FC6#,$*&$\"/+++++!***!#9F/FAF/F/F/F/*($\"/Be5lwn " 0 "" {MPLTEXT 1 0 82 "PL2:=pl ot(abs(lambda^2*c2_O2(t))^2,t=0..T,numpoints=500,color=blue): display( PL2);" }}{PARA 13 "" 1 "" {GLPLOT2D 1148 368 368 {PLOTDATA 2 "6%-%'CUR VESG6$7`jl7$$\"\"!F)F(7$$\"/#*>B2=cR!#:$\"/****4%)\\EU!#J7$$\"/:(oTK%) R(F-$\"/+g@fEjp!#H7$$\"/s-Y/'p7\"!#9$\"/**fDk;6?!#F7$$\"/Y[9Pk;:F:$\"/ +DhfEb@!#E7$$\"/%*Q')[Z/>F:$\"/\\U6DwC8!#D7$$\"/3fGS/kAF:$\"/]#[9]6D&F I7$$\"/HG[qNOEF:$\"/+*)[,og3O8!#B7$$\"/k%\\88+!QF:$\"/*4=**3H<$Fin7$$\"/&eW4^x 9%F:$\"/)[.fjVJ'Fin7$$\"/tNv;ARXF:$\"/KK?(o'z7!#A7$$\"/J2s'*HK\\F:$\"/ W*GygeW#Fio7$$\"/.\\dJ56`F:$\"/f4_//XVFio7$$\"/;F7I4bcF:$\"/0D%[f&eqFi o7$$\"/w,@-8kgF:$\"/jH([;F%))QHF^q7$$\"/(R=B'\\qrF:$\"/!3o[@IK%F^q7$$\"/#e@` 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"k=2" }{TEXT -1 36 " to the first- order probability for " }{TEXT 19 3 "k=1" }{TEXT -1 242 " we have more probability than what was removed from unity in the elastic channel. \+ This is why we may not want to use the actual probability calculation \+ for the elastic channel, but rather assume that it is 1 - sum (excitat ion probabilities)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 76 "We now check the Fourier transform interpretation of th e first-order result." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "FT:=unapply(abs(int(exp(I* omega*t)*Wmat(0,1),t=0..N*2*Pi/omega))^2,omega,N);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#FTGj+6$%&omegaG%\"NG6\"6$%)operatorG%&arrowGF),$*& $\"/++++++]!\")\"\"\")-%$absG6#*&,($\"$***\"\"!!\"\"*($F:F;F2-%$expG6# *&^#$\"/'zrI&=$G'!#8F29%F2F2-%$cosG6#,$*($\"/C(=7-pF'FFF2FGF29$F \+ " 0 "" {MPLTEXT 1 0 97 "plot([FT(omega,10),FT(omega,20),FT(omega,30)], 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`\"Fd^l7$Fggl$\"/-]U*>\"*p\"Fd^l7$$\"/BFp!zq/\"F-$\"/95Rmb'z\"Fd^l7$F \\hl$\"/myJ`0C=Fd^l7$Fahl$\"/)=H>'R-9Fd^l7$Ffhl$\"/IK7+*o4'Fjq7$Fa[o$ \"/sSV]/;GFjq7$F[il$\"/!4MoW6)pF-7$Fi[o$\"/j<))QK3[,)QFjq 7$Fjil$\"/Xi25SxcFjq7$F_jl$\"/YDxDq;\\Fjq7$Fdjl$\"/3-(3#[)y#Fjq7$Fijl$ \"/u3?l4T:Fjq7$F^[m$\"/:$e\"*y&>bF-7$Fc[m$\"/eC@!)=gWF17$Fh[m$\"/:F@z2 m%*F17$F]\\m$\"/x9*Gt\\:\"Fjq7$Fb\\m$\"/],>FaYAFjq7$Fg\\m$\"/A-WV&H/#F jq7$F\\]m$\"/6!==)o.@F-7$Fa]m$\"/b:U#[;\\#e(F-7$F`^m$\"/D`'R&zb%)F[\\l7$Fe^m$\"/5\\4#\\([fF-7$Fj^m$ \"/b(H9.P&zF-7$F__m$\"/#eK&*f&pXF-7$Fd_m$\"/j+Xk*4h#Fep7$Fj_m$\"/z(QU3 eM$F-7$F_`m$\"/NJ%z/n5&F-7$Fd`m$\"/*[%f`$*\\QF-7$Fi`m$\"/-TNq\")yRF1-F ^am6&F`amFdamFaamFdam-%+AXESLABELSG6$Q&omega6\"Q!Fh]q-%%VIEWG6$;F(Fi`m %(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Th e sharp turn-on and turn-off of the monochromatic pulse at time " } {TEXT 19 1 "0" }{TEXT -1 10 " and time " }{TEXT 19 7 "N*alpha" }{TEXT -1 332 " respectively causes a 'ringing' effect, namely the presence o f sidebands around the center frequency. Note how broad the signal is \+ for the number of cycles shown. The perturbation is very effective wit hin this window, i.e., transitions will take place if the transition f requency between initial and final state is within the peak." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "Think about th e consequences: for a long pulse of thousands of cycles (or more) an e ffective excitation becomes possible only for small (or negligible) de tuning from the resonance frequency." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 361 "Construct the Fourier transform of pulses containing more cycl es. Read up on windowing functions in a reference text on discrete (fa st) Fourier transforms to learn how they can suppress the ringing effe ct (e.g., Hanning window function, cf. digital oscilloscopes with FFT \+ function). Realistic laser pulses do have pulse shapes with gradual tu rn-on and turn-off." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 70 "Now we solve the close-coupling equations. We connect o nly the levels " }{TEXT 19 7 "k=0,1,2" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Wmat(0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"/b'=\"y1rq!#9\"\"\"-%$sinG6#,$*&$\"/+++++!***F'F (%\"tGF(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Wmat(1,2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,$*&$\"/+++++!***!#9\"\" \"%\"tGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "DE0:=I*diff( c0(t),t)=lambda*add(c||j(t)*Wmat(0,j)*exp(I/hbar*(En(j)-En(0))*t),j=0. .2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE0G/*&-%%diffG6$-%#c0G6#% \"tGF-\"\"\"^#F.F.,$**$\"/b'=\"y1rq!#:F.-%#c1GF,F.-%$sinG6#,$*&$\"/+++ ++!***!#9F.F-F.F.F.-%$expG6#*&F-F.F/F.F.F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 86 "DE1:=I*diff(c1(t),t)=lambda*add(c||j(t)*Wmat(1,j)*e xp(I/hbar*(En(j)-En(1))*t),j=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$DE1G/*&-%%diffG6$-%#c1G6#%\"tGF-\"\"\"^#F.F.,&**$\"/b'=\"y1rq!#:F. -%#c0GF,F.-%$sinG6#,$*&$\"/+++++!***!#9F.F-F.F.F.-%$expG6#*&^#!\"\"F.F -F.F.F.**$F.FDF.-%#c2GF,F.F7F.-F@6#*&F-F.F/F.F.F." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 86 "DE2:=I*diff(c2(t),t)=lambda*add(c||j(t)*Wmat (2,j)*exp(I/hbar*(En(j)-En(2))*t),j=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$DE2G/*&-%%diffG6$-%#c2G6#%\"tGF-\"\"\"^#F.F.,$**$F.! \"\"F.-%#c1GF,F.-%$sinG6#,$*&$\"/+++++!***!#9F.F-F.F.F.-%$expG6#*&^#F3 F.F-F.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "IC:=c0(0)=1,c 1(0)=0,c2(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6%/-%#c0G6#\" \"!\"\"\"/-%#c1GF)F*/-%#c2GF)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sol:=dsolve(\{DE0,DE1,DE2,IC\},numeric,output=listprocedure); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG7&/%\"tGj+6#F'6%%$resG%%dat aG%)solnprocG6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~Al l~rights~reserved.G6\"C&>8%`6$%$GetG%$SetGb6#%+thismoduleG6#%%DataGF06 $%$GetG%$SetGF0F0F0F0%+_m25403640G6#%%DataG>8&-_F3F>6#Q/soln_procedure F0@(-%'memberG6$9$7$Q%lastF0.%%lastGC$>8$-FD6#FO@$-%%typeG6$FT.%%listG O&FT6#\"\"\"3-FY6$FM%\"=G-FK6$-%$lhsG6#FM7$Q(initialF0.%(initialGC$@%- FY6$-%$rhsGFcoFen>FT-FD6#/Feo7$\"\"!-%#opG6#F\\p>FT-FD6#/Feo7$FjnF\\p@ $FXOFhn/FMQ(sysvarsF0O-FF6#F_qFMF0F0F06#\"+'yu.3#/-%#c0GF)j,6#%\"tG6'F +F,F-%)outpointG%&c0(t)G6#%inCopyright~(c)~2000~by~Waterloo~Maple~Inc. ~All~rights~reserved.GF0C)>%8_EnvDSNumericSaveDigitsG%'DigitsG>Fcr\"#9 @%/%-_EnvInFsolveG%%trueG>8'-&%&evalfG6#FbrFco>F[s-F^sFco>F3F4>FDFE@$4 -FY6$F[s.%(numericGC$@.-FK6$FM7+Q&startF0.%&startGQ'methodF0.%'methodG Q%leftF0Q&rightF0Q)leftdataF0Q*rightdataF0Q+enginedataF0C$@%-FK6$FM7$F `tFct>FT-FD6#-%(convertG6$FM.%'stringG>FT-FDFco@&-FY6$FT.%&arrayGO-%%e valG6$FTFjn0FTQ)procnameF0OFT-FK6$FM7$FOFeoC$Fgu@$FXO&FT6#\"\"#-FK6$FM 7$FPFfoC$F_uFiv3-FY6$F[sF^o-FK6$-Fbo6#F[sFdoC$@%-FY6$-F]pFhwFen>FT-FD6 #/Feo7$Fcp-Fep6#F]x>FT-FD6#/Feo7$F]wF]xFiv/F[sQ.solnprocedureF0O-F`v6# FD/F[sF_qOFaq@%09!%(unknownGO-.FcyFcoC$>8(-%(pointtoG6#&-FF6#Q0soln_pr oceduresF0F\\wO-.FjyFcoZ%C$>FT-FDFhwF[wF0YF0F0F0F06#\"+'yu.3#F0/-%#c1G F)j,6#%\"tG6'F+F,F-F\\r%&c1(t)GF^rF0C)>FbrFcr>FcrFer@%Fgr>F[sF\\s>F[sF as>F3F4>FDFE@$FesC$@.F\\tC$@%F\\u>FTF`u>FTFhu@&FjuOF_vFbvOFTFevC$Fb\\l @$FXO&FT6#\"\"$F^wC$Fa\\lFg\\lFbwC$@%F[x>FTF_x>FT-FD6#/Feo7$F[]lF]xFg \\lFjxOF]yF_yOFaq@%FbyOFfyC$>Fjy-F\\z6#&F_zFj\\lOFczZ%C$>FTFhzFi\\lF0Y F0F0F0F06#\"+'yu.3#F0/-%#c2GF)j,6#%\"tG6'F+F,F-F\\r%&c2(t)GF^rF0C)>Fbr Fcr>FcrFer@%Fgr>F[sF\\s>F[sFas>F3F4>FDFE@$FesC$@.F\\tC$@%F\\u>FTF`u>FT Fhu@&FjuOF_vFbvOFTFevC$F[`l@$FXO&FT6#\"\"%F^wC$Fj_lF``lFbwC$@%F[x>FTF_ x>FT-FD6#/Feo7$Fd`lF]xF``lFjxOF]yF_yOFaq@%FbyOFfyC$>Fjy-F\\z6#&F_zFc`l OFczZ%C$>FTFhzFb`lF0YF0F0F0F06#\"+'yu.3#F0" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 62 "C0:=eval(c0(t),sol): C1:=eval(c1(t),sol): C2:=eval( c2(t),sol):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "PL4:=plot([ abs(C1(t))^2,abs(C2(t))^2],t=0..T,color=[magenta,black]): display(PL2, PL1,PL4,title=\"Comparison of level populations P1(t)[magenta=close-co upling; red=TDPT] and P2(t)\");" }}{PARA 13 "" 1 "" {GLPLOT2D 1076 409 409 {PLOTDATA 2 "6)-%'CURVESG6$7`jl7$$\"\"!F)F(7$$\"/#*>B2=cR!#:$ \"/****4%)\\EU!#J7$$\"/:(oTK%)R(F-$\"/+g@fEjp!#H7$$\"/s-Y/'p7\"!#9$\"/ **fDk;6?!#F7$$\"/Y[9Pk;:F:$\"/+DhfEb@!#E7$$\"/%*Q')[Z/>F:$\"/\\U6DwC8! 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F-7$F^i[l$\"/.80ugbLF-7$Fhi[l$\"/^nC8<1QF-7$F]j[l$\"/0s+,5]RF-7$Fbj[l$ \"/,K=W]GSF-7$Fgj[l$\"/+1Dl\"f0%F-7$F\\[\\l$\"/_/-,7fSF-7$$\"/(eAv([#e \"F`_p$\"/z(>(G$yF-7$F]_\\l$\"/X*oS\\y%yF-7$$\"/!RX:$fw=F`_p$\"/%of.sG&yF-7$Faht$\" /8\\e;6`yF--Ffht6&FhhtF)F)F)-%+AXESLABELSG6%Q\"t6\"Q!F`^]l-%%FONTG6#%( DEFAULTG-%&TITLEG6#Q\\pComparison~of~level~populations~P1(t)[magenta=c lose-coupling;~red=TDPT]~and~P2(t)F`^]l-%%VIEWG6$;F($\"/&)eMC%o)=F`_pF e^]l" 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 86 "The three-channel close-coupling result is not exact, as it neglec ts couplings to the " }{TEXT 19 3 "k=3" }{TEXT -1 57 " state. Given th at the couplings proceed in steps, i.e., " }{TEXT 19 3 "k=1" }{TEXT -1 19 " is populated from " }{TEXT 19 3 "k=0" }{TEXT -1 2 "; " }{TEXT 19 4 "k=2 " }{TEXT -1 18 "is populated from " }{TEXT 19 3 "k=1" } {TEXT -1 70 ", etc. we expect this to affect the results at the latest times shown." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "This feature of step-wise excitation also explains why t he discrepancy between the first-order result for " }{TEXT 19 5 "P1(t) " }{TEXT -1 67 " overestimates the close-coupling result by about the \+ level of the " }{TEXT 19 5 "P2(t)" }{TEXT -1 33 " calculated in second -order TDPT." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Note how the 2nd order result for " }{TEXT 19 5 "P2(t)" }{TEXT -1 85 " agrees with the close-coupling calculation at times when the f irst-order answer for " }{TEXT 19 5 "P1(t)" }{TEXT -1 67 " is overesti mating the the corresponding TDSE result significantly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 3:" }} {PARA 0 "" 0 "" {TEXT -1 66 "Extend the close-coupling calculation to \+ include couplings to the " }{TEXT 19 3 "k=3" }{TEXT -1 121 " level. At what time do these populations become appreciable? What is the implic ation for the accuracy of the 3-channel (" }{TEXT 19 7 "k=0,1,2" } {TEXT -1 42 ") close-coupling calculations shown above?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "The philosophy in n th order TDPT is: use the probability amplitudes in previous order for the other levels to obtain the k-level correction at the order to be \+ computed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "A few numerical comparisons:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "abs(C0(T))^2+abs(C1(T))^2+abs(C2(T))^2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"/Z_g?++5!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalf(eval(P_0k(1),s=T)),abs(C1(T))^2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$$\"/-))RZjXW!#9$\"/0!)y\\R%z#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(eval(P_0k(1),s=T/2)),abs(C1(T /2))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"/UM:MV66!#9$\"/,5(fhj$** !#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(eval(P_0k(1),s =T/4)),abs(C1(T/4))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"/'*Hpa]1H !#:$\"/:kb/1BGF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "This raises t he question:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "to what order do we need to go in order to obtain agreement be tween the TDPT approach and the integration of the close-coupling equa tions not just at the beginning?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 61 "Let us try the higher-order result for th e excitation of the " }{TEXT 19 3 "k=1" }{TEXT -1 7 " state." }}{PARA 0 "" 0 "" {TEXT -1 44 "The scheme is to connect neighboring levels:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "0->1 hap pens at O1" }}{PARA 0 "" 0 "" {TEXT -1 126 "1->2 and 1->0 at O2. No c ontribution 0->1 at this level because the 1st order contribution to t he c1(t) coefficient vanishes." }}{PARA 0 "" 0 "" {TEXT -1 56 "So let \+ us calculate the 3rd order contribution to c1(t)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "c1_O3:=una pply(-I/hbar*(int(exp(I*(En(1)-En(0))*t)*Wmat(0,1)*c0_O2(t),t=0..s)+in t(exp(I*(En(1)-En(2))*t)*Wmat(2,1)*c2_O2(t),t=0..s)),s);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%&c1_O3Gj+6#%\"sG6\"6$%)operatorG%&arrowGF(*&^# !\"\"\"\"\",:*($\"/Ohy,+:W!\"'F/-%$expG6#*&^#$F/\"\"!F/9$F/F/-%$cosG6# ,$*&$\"/+++++!***!#9F/F@#!\"*F/F5F/-%$sinG6#,$*& $\"/+++++(*H!#8F/F@#FIF/F5F/-F>FLF/F/*(^#$!/3\"HY 0sT%F4F/-F66#*&^#$F.F;F/FWF4F/F5F/FinF/F/ **$\"/Fd;,%QU%FIF/F5F/FinF/F6#,$*&$\"/+++++)* >FQF/FWFIF/F@#FIF/-FKFeoF/F/$ \"/3JRc?K))F4F/*($\"/zWd#)y7WF4F/FZF/F=F/F.**^#$\"/iXwiT>WFIF/F5F/F=F/ F " 0 "" {MPLTEXT 1 0 9 "c1_O3( 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!(7(ph!\"*$\")Q&4^\"F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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At late times the third-orde r correction overcompensates the large deviation in the first-order re sult." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 201 "One can view the TDPT scheme as a successive approximation scheme tha t integrates the system of ordinary differential equations (the CC equ ations form of the TDSE) in a successive approximation scheme." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 11 "Exercise 4:" }}{PARA 0 "" 0 "" {TEXT -1 19 "The calculation of " }{TEXT 19 8 " c1_O3(t)" }{TEXT -1 566 " represents the complete pattern of the TDPT \+ scheme for the present problem. Nearest neighbors are connected by the perturbation. One uses the coefficients from the previous approximati on (not the complete coefficient, just the contribution from that orde r!) multiplies it with the appropriate coupling matrix and computes th e Fourier transform for fixed transition frequency.The two amplitudes \+ from the lower-lying and higher-lying neighboring energy levels are ad ded coherently to form the overall probability amplitude. Use this pat tern to compute the population " }{TEXT 19 5 "P1(t)" }{TEXT -1 65 " at the next order for which there is a non-trivial contribution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "We proceed with an improved calculation of the \+ probability to remain in the (" }{TEXT 19 3 "k=0" }{TEXT -1 94 ") grou nd state. In this case there is only one contribution, as the connecti on is only to the " }{TEXT 19 3 "k=1" }{TEXT -1 39 " state. We have co ntributions at order " }{TEXT 19 8 "lambda^2" }{TEXT -1 5 " and " } {TEXT 19 8 "lambda^4" }{TEXT -1 52 " which can be calculated from the \+ known results for " }{TEXT 19 2 "c1" }{TEXT -1 53 " at orders O1 and O 3. The O2 result is known already:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "c0_O2(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!)i1%) \\!\"*$\")&3@P\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "c0_O4 :=unapply(-I/hbar*int(exp(I*(En(0)-En(1))*t)*Wmat(1,0)*c1_O3(t),t=0..s ),s);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&c0_O4Gj+6#%\"sG6\"6$%)oper atorG%&arrowGF(*&^#!\"\"\"\"\",F^#$\"/K7`%*o/y!\"%F/*(^#$\"/%****\\(o/ yF4F/-%$expG6#*&^#$!\"#\"\"!F/9$F/F/-%$cosG6#,$*&$\"/+++++)*>!#8F/FAF/ F/F/F/*($\"/,+++]i:!\"$F/-F:6#*&^#$F.F@F/FAF/F/-%$sinG6#,$*&$\"/+++++! ***!#9F/FAF/F/F/F/*($\"/8v$f$o/yF4F/F9F/-FTFDF/F.*&^#$!/&y4++D\"y!\"(F /FBF/F/**$\"/\"f$ozrh:!\"'F/FNF/-FCFUF/FAF/F.*&$\"/d>++]i:FaoF/FAF/F/* ($\"/9&Qv1/!Q!#BF/FNF/-FT6#,$*&$\"/+++++(*HFIF/FAF/F/F/F/*&$\"/(*H@.K? yF]oF/FhnF/F.*&$\"/%[uiOq&>!#5F/-FT6#,$*&$\"/+++++'*RFIF/FAF/F/F/F/*&^ #$\"/i%pgPq&>FfpF/-FCFhpF/F/*(^#$!/SGj$=.#yFfpF/FNF/-FCF[pF/F/*(^#$\"/ Rz4l9GyFfpF/FAF/FhnF/F/*&^#$!/TGj$=.#yFfpF/)FA\"\"#F/F/**^#$!/)3'\\7Gj :FaoF/FNF/FSF/FAF/F/*(^#$!/******\\Pf:FMF/FNF/FboF/F/*&^#$!/jzT)*e3yF] oF/F9F/F/*($\"/VGj$=.#yFfpF/FAF/FBF/F.F/F(F(F(6#F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "c0_O4(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$\"&x:\"!\"($!%vrF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 " plot([abs(C0(t))^2,abs(1+lambda^2*c0_O2(t))^2,abs(1+lambda^2*c0_O2(t)+ lambda^4*c0_O4(t))^2],t=0..T,numpoints=500,color=[magenta,red,brown],t itle=\"P0(t): CC[magenta], TDPT-O2[red], TDPT-O4[brown]\");" }}{PARA 13 "" 1 "" {GLPLOT2D 1296 341 341 {PLOTDATA 2 "6(-%'CURVESG6$7`jl7$$\" \"!F)$\"\"\"F)7$$\"/#*>B2=cR!#:$\"/yRs********!#97$$\"/:(oTK%)R(F/$\"/ Y$)H'*******F27$$\"/s-Y/'p7\"F2$\"/FQ$*z******F27$$\"/Y[9Pk;:F2$\"/hPL M******F27$$\"/%*Q')[Z/>F2$\"/-_CP)*****F27$$\"/3fGS/kAF2$\"/8&ymn**** *F27$$\"/HG[qNOEF2$\"/LjB3%*****F27$$\"/w%eq.9-$F2$\"//H+$)*)****F27$$ \"/$)*Q^:_S$F2$\"/a:Om$)****F27$$\"/k%\\88+!QF2$\"/Mw(4[(****F27$$\"/& eW4^x9%F2$\"/Gv(fW'****F27$$\"/tNv;ARXF2$\"/]l9T\\****F27$$\"/J2s'*HK \\F2$\"/8?%p+$****F27$$\"/.\\dJ56`F2$\"/Pj(*z1****F27$$\"/;F7I4bcF2$\" /H%*3A\"))***F27$$\"/w,@-8kgF2$\"/o![;Y%)***F27$$\"/XK!RQ1T'F2$\"/+5gz 2)***F27$$\"/@Hm(4P\"oF2$\"/!pN*fd(***F27$$\"/(R=B'\\qrF2$\"/!3m.gq*** F27$$\"/#e@`Z>c(F2$\"//o]/T'***F27$$\"/-BS`@2z!o)F2$\"/9-%=m%*F2$\"/G,_)3=***F27$$\"/(G.aOX\")*F2$ \"/+33#y1***F27$$\"/0nIa2>5!#8$\"/csW&[$*)**F27$$\"/1B2C%z0\"F]t$\"/#y C\"R&y)**F27$$\"/64hh'f4\"F]t$\"/(**F27$$\"/^ Vz0b)R\"F]t$\"/Wr\"f4%p**F27$$\"/oG2J)eV\"F]t$\"/G*HzOo'**F27$$\"/!\\T !o'\\Z\"F]t$\"/DTiC/k**F27$$\"/[V\\!4E^\"F]t$\"/c_F+Eh**F27$$\"/'H?o.6 b\"F]t$\"/0,r'H$e**F27$$\"/4jf&z#*e\"F]t$\"/))=DrMb**F27$$\"/KNE#fVi\" F]t$\"/\"RD\"pa_**F27$$\"/a:$[kXm\"F]t$\"/f3?pF\\**F27$$\"/'HTaC0q\"F] t$\"/:v.iIY**F27$$\"/rRpa')QF]t$\"/8iFz2F**F27$$\"/44 -l1o>F]t$\"/d([SSP#**F27$$\"/f13Z^-?F]t$\"/**e0h*3#**F27$$\"/)3O>Z#F]t$\"/I@ T?C1**F27$$\"/at$z$oIAF]t$\"/uBRsZ.**F27$$\"/!3B?J!oAF]t$\"/gV%3^4!**F 27$$\"/i>\"fPtI#F]t$\"/5%el:%)*)*F27$$\"/HM#F]t$\"/%p)=eB'*)*F27$ $\"/;)*HQC#Q#F]t$\"/:9^.'R*)*F27$$\"/rSS)4*>CF]t$\"/E'fX<>*)*F27$$\"/ \"42<%[dCF]t$\"/Y(Gv;+*)*F27$$\"/@M/2t'\\#F]t$\"/kc%3\"=)))*F27$$\"/Pc %>yG`#F]t$\"/tj^wi'))*F27$$\"/f/p\")*)pDF]t$\"/Hg#3u^))*F27$$\"/Gm>Ex5 EF]t$\"/Y#*R-t$))*F27$$\"/[n'3\"yZEF]t$\"/r?Csc#))*F27$$\"/4Ew\"Hco#F] t$\"/(*oso^\"))*F27$$\"/5Z3(>Ts#F]t$\"/51^))e!))*F27$$\"/Jf(***\\fFF]t $\"/e*y(e&)z)*F27$$\"/v4SU@(z#F]t$\"/>_7_>z)*F27$$\"/kL*RTY$GF]t$\"/Kb 3bly)*F27$$\"/3&f#onuGF]t$\"/*ez\\'>y)*F27$$\"/$3cN1+\"HF]t$\"/QuiS)y( )*F27$$\"/Hv)Hp2&HF]t$\"/:'QW>w()*F27$$\"/&R-$F]t$\"/zq'o`t()*F27$$\"/J-'o!4jIF]t$\"/R'*fPHx)*F27$$\"/f9 SxQ-JF]t$\"/?&)3.Fx)*F27$$\"/:zSA9QJF]t$\"/j%zMms()*F27$$\"/yEK>6wJF]t $\"/]zpWEx)*F27$$\"/R:Sm;8KF]t$\"/Egm)[s()*F27$$\"/HUMVL`KF]t$\"/si%=( >x)*F27$$\"/Kyd04)G$F]t$\"/Q7#e2r()*F27$$\"/>)QM4!GLF]t$\"/^^VO$p()*F2 7$$\"/YT,^`lLF]t$\"/h&[)>ow)*F27$$\"/J0,%)p-MF]t$\"/:D$QIj()*F27$$\"/T Pk7*)RMF]t$\"/Ivq!fe()*F27$$\"/8oIc#yZ$F]t$\"/tz9vBv)*F27$$\"/yYBa.=NF ]t$\"/;IdNSu)*F27$$\"/V(f'Q!\\b$F]t$\"/(G)4HYt)*F27$$\"/<'Gv(H\"f$F]t$ \"/FY(=`B()*F27$$\"/q(*=A?IOF]t$\"/#f68^4()*F27$$\"/+Z$*>-pOF]t$\"/WI? 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ht$\"))eqk\"F_htFad`l-%+AXESLABELSG6$Q\"t6\"Q!Fgd`l-%&TITLEG6#QQP0(t): ~CC[magenta],~TDPT-O2[red],~TDPT-O4[brown]Fgd`l-%%VIEWG6$;F($\"/&)eMC% o)=Fd^p%(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" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 319 "We can observe the impressive agreement between the CC result and the O4-TDPT result for the elastic channel. If the system \+ were evolved for a longer time, or for stronger coupling lambda we sho uld observe that TDPT at high order will converge against the CC resul t with more equations coupled than in the present case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 217 "One should keep in \+ mind that the coupling scheme is particularly simple for the harmonic \+ oscillator. For anharmonic potentials direct coupling to more excited \+ states is possible by the time-dependent linear potential." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 13 "Mini-project:" } }{PARA 0 "" 0 "" {TEXT -1 388 "Explore what happens to the above resul ts when the strength parameter lambda is increased (e.g., doubled, etc .). Be careful in your interpretation of results. Note when the calcul ated probability expression exceeds unity and understand what this mea ns. Extend the CC equations to sufficient matrix dimension to guarante e converged results against which the TDPT calculations are compared. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "71" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }