{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 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 "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 Output" -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 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 46 "The matrix representatio n of quantum mechanics" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We use the matrix representation of quantum mechani cs to approximate the energy spectrum of an anharmonic oscillator." }} {PARA 0 "" 0 "" {TEXT -1 119 "We make use of the particle-in-a-box bas is. The box size has to be chosen sufficiently large so that the trunc ation of " }{TEXT 264 1 "x" }{TEXT -1 39 "-space to a finite domain is justified." }}{PARA 0 "" 0 "" {TEXT -1 112 "An alternative would have been to choose a harmonic oscillator basis (as done in Quantum Mechan ics using Maple)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 206 "We construct the eigenfunctions using the new LinearAlge bra package of Maple 6. At the end we compare the approximate eigenfun ction with a numerical solution to the differential-equation eigenvalu e problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 279 "We solve the problem by calculating matrix elements in the coo rdinate representation. Note, however, that a worksheet exists to carr y out the calculations based on the commutation relations of quantum m echanics. These worksheets are commut1.mws (Maple5), and define.mws (M aple6)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "We start with the basis. The box size will put limits on the accu racy of the higher-lying eigenvalues/eigenfunctions. We also choose a \+ truncation size for the matrix eigenvalue problem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "restart; Digits:=15: with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "X:=8; N:=20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"NG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "uB:=n->if is( n,even) then sqrt(1/X)*sin(n*Pi*x/(2*X)) else sqrt(1/X)*cos(n*Pi*x/(2* X)) end if:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot([uB(1), uB(2),uB(3),uB(4)],x=-X..X,color=[red,blue,green,black]);" }}{PARA 13 "" 1 "" {GLPLOT2D 691 218 218 {PLOTDATA 2 "6(-%'CURVESG6$7S7$$!\")\"\" !$F*F*7$$!0LLL`X7l(!#9$\"0?pWTj\">C!#;7$$!0nmm8'zZtF/$\"0X=WSP_^%F27$$ !0LLLbNl+(F/$\"03YEu!)H&oF27$$!0LLL(G,jmF/$\"0&*G#3J8v\"*F27$$!0nmm*G7 @jF/$\"0#G%))\\$[W6!#:7$$!0MLL7ZT+'F/$\"0%>08+L]8FG7$$!0+++2Pfn&F/$\"0 IYAB_zb\"FG7$$!0MLLw,lL&F/$\"0W&)fwhew\"FG7$$!0+++.b\")*\\F/$\"0^&*e>- `'>FG7$$!0nmm#Q7]YF/$\"0MN*4iRh@FG7$$!0MLLvxNM%F/$\"0\"\\rFzyDBFG7$$!0 +++[z%)*RF/$\"0)33siu+DFG7$$!0+++?k>l$F/$\"0XSM*>F/$\"03d')z_\"oKFG7$$!0nm m)p*)y;F/$\"0&Q;8[;XLFG7$$!0,++r:QL\"F/$\"07=W#o(\\T$FG7$$!0,++t;_+\"F /$\"0=YKG\"*oY$FG7$$!/nmm;eBmF/$\"0#4'y1wc]$FG7$$!/nmm(p]Z$F/$\"00V4(o IFNFG7$$!-MLLV(*yF/$\"03q_lHb`$FG7$$\"/mmmVh[MF/$\"0+N)4;VFNFG7$$\"/** ***p!R>lF/$\"0s]Ev1m]$FG7$$\"/mmmK\"f$)*F/$\"0_l#oQ!)pMFG7$$\"0*****f0 AE8F/$\"0Q7p9QjT$FG7$$\"0*****>kTh;F/$\"0][#eL2\\LFG7$$\"0*****\\ct&)> F/$\"0o&y[S=qKFG7$$\"/++go$eM#!#8$\"0y#H*Qvq;$FG7$$\"/LL8QSpEFbu$\"0v& 4s;\"41$FG7$$\"/+++1)[,$Fbu$\"0ly%ft$R$HFG7$$\"/nm;R$zK$Fbu$\"0S>S'y?2 GFG7$$\"/++!)Q=qOFbu$\"0Sx\\gWll#FG7$$\"/LLBW@#*RFbu$\"0A)Q8)=Q]#FG7$$ \"/++I\"H)GVFbu$\"0k:3W*[LBFG7$$\"/LLL:$zl%Fbu$\"0qqnM/r:#FG7$$\"/++]7 Z-]Fbu$\"0y4Iu5G'>FG7$$\"/nmYRIM`Fbu$\"0ydFR$=njFbu$\"0UcIE&fX6 FG7$$\"/LL8p&Qn'Fbu$\"0)Q$)o?T-\"*F27$$\"/nmE/'3*pFbu$\"0-$z'=,(fpF27$ $\"/++!H_)GtFbu$\"0e7pGPck%F27$$\"/++ION_wFbu$\"0kHP(*)[6CF27$$\"\")F* F+-%'COLOURG6&%$RGBG$\"*++++\"F)F+F+-F$6$7enF'7$F-$!0qO[I()p#[F27$F4$! 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What are the eigenenergies? We have \+ kinetic energy only, and evaluate it only over the allowed range; the \+ result agrees with the textbook formula. The kinetic energy matrix wil l be diagonal in our basis, as the basis functions are eigenfunctions \+ of the kinetic energy operator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Tkin:=psi->int(psi*(-1/2*diff(psi,x$2)),x=-X..X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%TkinGR6#%$psiG6\"6$%)operatorG%&arrowGF(-%$i ntG6$,$*&9$\"\"\"-%%diffG6$F1-%\"$G6$%\"xG\"\"#F2#!\"\"F:/F9;,$%\"XGF< F@F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "[Tkin(uB(1)),Tk in(uB(2)),Tkin(uB(3)),Tkin(uB(4))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7&,$*$)%#PiG\"\"#\"\"\"#F)\"$7&,$F%#F)\"$G\",$F%#\"\"*F+,$F%#F)\"#K" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(n^2*Pi^2/8/X^2,n=1..4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&,$*$)%#PiG\"\"#\"\"\"#F(\"$7&,$F $#F(\"$G\",$F$#\"\"*F*,$F$#F(\"#K" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Now let us define our potential of interest:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "V:=x^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"VG*$)%\"xG\"\"%\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "We de fine matrix elements where, again, space is restricted to a finite int erval. This is imposed by our choice of basis! We allow two different \+ functions in bra and ket." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Vpot:=(phi,psi)->int(expand(phi*psi*V),x=-X..X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VpotGR6$%$phiG%$psiG6\"6$%)operatorG%&arrowGF)-%$ intG6$-%'expandG6#*(9$\"\"\"9%F5%\"VGF5/%\"xG;,$%\"XG!\"\"F " 0 "" {MPLTEXT 1 0 14 "HM:=Matr ix(N):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "We make use of the sym metry of the hamiltonian matrix: it allows to save almost a factor of \+ 2 in computation time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 " for i from 1 to N do: for j from 1 to i do: HM[i,j]:=Vpot(uB(i),uB(j)) ; if i=j then HM[i,j]:=HM[i,j]+ Tkin(uB(i)); fi: od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for i from 1 to N do: for j from i+ 1 to N do: HM[i,j]:=HM[j,i]: od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "print(SubMatrix(HM,1..4,1..4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")w\\(\\\"-%'MATRIXG6#7&7&,&*&,(*$)%#PiG\" \"%\"\"\"F3\"$?\"F3*&\"#?F3)F1\"\"#F3!\"\"F3*$F0F3F9#\"%'4%\"\"&*&#F3 \"$7&F3F7F3F3\"\"!,$*&,&!#:F3*&F8F3F7F3F3F3*$F0F3F9!%WhFA7&FA,&*&,(\"# :F3*&\"#5F3F7F3F9*&F8F3F0F3F3F3*$F0F3F9#\"%[?F=*&#F3\"$G\"F3F7F3F3FA,$ *&,&!#?F3*&\"\"$F3F7F3F3F3*$F0F3F9#!'s58\"#F7&FBFA,&*&,(*$F7F3!#g\"#SF 3*&FjnF3F0F3F3F3*$F0F3F9#F<\"$N\"*&#\"\"*F@F3F7F3F3FA7&FAFWFA,&*&,(F_o !#S*&\"#KF3F0F3F3FMF3F3*$F0F3F9#FVF=*&#F3F_pF3F7F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 251 "All integrals were calculated in closed form. \+ The matrix has a sparse structure: the chosen potential respects parit y symmetry, and the basis consists of even/odd functions. Therefore, o ne could solve the problem of the even and odd eigenfunctions of " } {TEXT 259 1 "V" }{TEXT -1 100 " separately. We do keep it general, so \+ that one may insert a non-symmetric potential function above." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "We canno t expect the matrix diagonalization to be carried out symbolically. Th erefore, we switch to floating-point evaluation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "HMf:=map(evalf,HM):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evals:=Eigenvalues(HMf, output='list'):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ev_s:=sort(map(Re,evals));" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%%ev_sG76$\"3b3+g2WX3n!#=$\"3<8%oza,)4C!#<$\"3D7E8t/ aA`F+$\"3%[*)HAF%e%f*F+$\"3_V'QMk_41#!#;$\"3cu-*o7Jj@$F2$\"3/y9H7_!)) \\'F2$\"3%oGL^&[hp))F2$\"35;+h]x#)Q;!#:$\"3vy%pcoV].#F;$\"3lo:#RG!R0NF ;$\"3C(***zT#H`2%F;$\"3o*o#4d([Cm'F;$\"3)yHsu4/SQ(F;$\"3+\"***GG\"*>g6 !#9$\"3#p_5M'RDS7FH$\"3')G*y=Od*))=FH$\"3-8D?0`Cj>FH$\"3'eF:i4'HFH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 173 "Carry out matrix diagonalizations with submatrices of the N-by-N Hamiltonian matrix, and observe the behavio ur of the low-lying eigenvalues as a function of truncation size." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 " Now the eigenfunctions. We need a sorting procedure to arrange the res ult from the eigenvector calculation in proper order." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "VE:=Eigenvectors(HMf,output='list') :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Vp:=[seq([Re(VE[i][1]) ,VE[i][2],map(Re,VE[i][3])],i=1..nops(VE))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 344 "Min:=proc(x,y); if type(x,numeric) and type(y,n umeric) then if x<=y then RETURN(true): else RETURN(false): fi; elif t ype(x,list) and type(y,list) and type(x[1],numeric) and type(y[1],nume ric) then if x[1]<=y[1] then RETURN(true): else RETURN(false): fi; eli f convert(x,string)<=convert(y,string) then RETURN(true): else RETURN( false): fi: end:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "VEs:= sort(Vp,Min):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 252 "Suppose that \+ we would like to see the eigenfunctions corresponding to the four lowe st-lying eigenvalues. The eigenvector for a given eigenvalue contains \+ the expansion coefficients for the expansion of the eigenstate in term s of the chosen basis states." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 4 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "psi0 :=add(VEs[i][3][j]*uB(j),j=1..N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "No:=1/sqrt(int(expand(psi0^2),x=-X..X));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "phi_a[i]:=add(No*VEs[i][3][j]*uB(j),j=1..N): od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([seq(phi_a[i],i=1..4)], x=-5..5,color=[red,blue,green,black]);" }}{PARA 13 "" 1 "" {GLPLOT2D 695 232 232 {PLOTDATA 2 "6(-%'CURVESG6$7[p7$$!\"&\"\"!$!0!=98>UCq!#>7$ $!0LL$e%G?y%!#9$\"0\")**G'Hu)z'F-7$$!0n;aesBf%F1$\"08)p)ezU%>!#=7$$!0L L3s%3zVF1$\"0P'osV7JAF97$$!0MLe/$QkTF1$\"0N)*=%ftJZF-7$$!0n;/\"=q]RF1$ !0x3*pv')3FF97$$!0M$3_>f_PF1$!0u&\\2vf8\\F97$$!0+](o1YZNF1$!0*>ebqvIRF 97$$!0M$3-OJNLF1$\"0Q1!\\p[K;F97$$!0+]P*o%Q7$F1$\"0I?BTw,6*F97$$!0nm;R 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Fer7$Fhem$!0k4`-N,c$F`q7$Fc`l$\"0dn)\\b<\"4#F`q7$Faao$\"0xCX?Gv6%F`q7$ F[bo$\"0y_h*zANbF`q7$$\"0*\\PMTiEEF1$\"0E%R2$*RGgF`q7$F`bo$\"0PJ*RzI'Q 'F`q7$$\"0m\"H#=?#zEF1$\"0#)RsdGjh'F`q7$Fh`l$\"0k+kf3ls'F`q7$$\"0U5:LH 7t#F1$\"0]Z\\IUms'F`q7$$\"0$3xc/%pv#F1$\"0]5&GO'*GmF`q7$$\"0DJ?e^Ey#F1 $\"0-)*))f;BW'F`q7$Fhbo$\"0-EnQ,e<'F`q7$$\"0\\7y&\\yfGF1$\"0WM,q>.W&F` q7$F]al$\"0y-m(\\8(\\%F`q7$Faco$\"0*)*GU\"o*o@F`q7$Fbal$!/v'R(zy([\"F` q7$F^do$!0Tw(e'G'R>F`q7$Fgal$!0:fy*Q`WIF`q7$$\"0kTN\"H'pQ$F1$!09:Hf^AJ $F`q7$F[eo$!0o%f2\">CQ$F`q7$$\"0k\"H2)3I\\$F1$!0r4Lw(QpKF`q7$F\\bl$!0[ uMxd^*HF`q7$$\"0)\\PM&*>^OF1$!0Y9'**)\\!)3#F`q7$Fabl$!/$*o4%=.;*F`q7$F feo$\"/Utbw6x;F`q7$Ffbl$\"0*R'H(='e2\"F`q7$F^fo$\"0gEOw&>VF`q7$$\"0)*\\(oTAqUF1$\"0xS?G'eB%\\L7F` q7$Fecl$!/@\\::,&z\"F`q7$$\"0)****\\oi\"o%F1$!/t)f@H3`(F`q7$Fjcl$!0)HU I25A6F`q7$$\"0*\\P40O\"*[F1$!00'Qu#G+C\"F`q7$F_dl$!0R%R%)38y5F`q-Fddl6 &FfdlF*F*F*-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(F_dl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The fo ur functions should have 0,1,2,3 nodes. The additional nodes at large \+ " }{TEXT 261 1 "x" }{TEXT -1 44 " are clearly an artefact of the calcu lation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Let us check the accuracy of these approximate eigenfunct ions one by one:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ev_s[1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3b3+g2WX3n!#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "We can construct a shooting-method solution: we will use " }{TEXT 19 6 "dsolve" }{TEXT -1 60 " in numeric mode to int egrate out the Schroedinger equation." }}{PARA 0 "" 0 "" {TEXT -1 59 " For the even-parity state we choose boundary conditions as " }{TEXT 19 17 "u(0)=1, D(u)(0)=0" }{TEXT -1 32 "; and for the odd-parity state s " }{TEXT 19 17 "u(0)=0, D(u)(0)=1" }}{PARA 0 "" 0 "" {TEXT -1 128 "W e use the energy as a trial value to ensure that the wavefunction vani shes at the 'boundary' (where the potential grows large)." }}{PARA 0 " " 0 "" {TEXT -1 153 "This 'boundary' value is state-dependent. One ite rates the procedure by hand. One can also write a bisection algorithm \+ to automate the eigenvalue search." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "IC:=u(0)=1,D(u)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6$/-%\"uG6#\"\"!\"\"\"/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E_t:=0.668;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "SE:=-1/2*diff(u(x),x$2)+(V-E_t)*u(x)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sol:=dsolve(\{SE,IC\},u(x),numeric,output=listprocedu re):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u_s:=subs(sol,u(x)):" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E_tG$\"$o'!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SEG/,&-%%diffG6$-%\"uG6#%\"xG-%\"$G6$F-\"\"##!\"\"F1 *&,&*$)F-\"\"%\"\"\"F9$\"$o'!\"$F3F9F*F9F9\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "P1:=plot('u_s(x)',x=0..2.5): plots[display](P1 );" }}{PARA 13 "" 1 "" {GLPLOT2D 693 187 187 {PLOTDATA 2 "6%-%'CURVESG 6$7S7$$\"\"!F)$\"\"\"F)7$$\"0mm;a)G\\a!#;$\"0oUV[q,)**!#:7$$\"0Lek`o!> 5F2$\"0&*>'o!42$**F27$$\"0n;z>)G_:F2$\"0$G$pRz%R)*F27$$\"0n;aQU!*3#F2$ \"0F%4/k%*4(*F27$$\"0LeRZXKi#F2$\"05L+*Q/W&*F27$$\"0m\"z>,_=JF2$\"0.-` \"H&zN*F27$$\"0+D\"G$[8j$F2$\"08iD^G etXF27$$\"0]7`H\"fT5!#9$\"0na=VNp7%F27$$\"0]Pf)[$H4\"Fjq$\"0v!)4W&)Gr$ F27$$\"0Lek`1l9\"Fjq$\"0ei3XEnH$F27$$\"0$e*[.-d>\"Fjq$\"0<)pOW5KHF27$$ \"0nTg-m([7Fjq$\"09tIDI2c#F27$$\"0F27$$\"0n\"zWho.9Fjq$\"0cs-5_'G;F27$$\"0++D'>A d9Fjq$\"0,H)\\Rvj8F27$$\"0+DcJ'f4:Fjq$\"02'GE.hL6F27$$\"0]7`>r-c\"Fjq$ \"0&H\"R'Q!pP*F/7$$\"0+v$4q`;;Fjq$\"0&ycAIF(\\(F/7$$\"0LLeM%4n;Fjq$\"0 T^)\\18egF/7$$\"0+]P4v5s\"Fjq$\"0Fr=ct?w%F/7$$\"0Fjq$\"0tK,Zycm\"F/7$$\"0L$3-=!y(>Fjq$\"0CL@-2HB\"F/7$$\"0]7 G8O;.#Fjq$\"0u!H)z3\">))!#<7$$\"0nm;*\\[$3#Fjq$\"0!Q#*fwiLiF\\x7$$\"0n T&Qz]O@Fjq$\"0r[V:J+A%F\\x7$$\"0$ekG=4*=#Fjq$\"0By6#Hb(p#F\\x7$$\"0++D '4TPAFjq$\"0%*>q8ycf\"F\\x7$$\"0L3F9!z#H#Fjq$\"0*ePVFK1b!#=7$$\"0nm;%> KUBFjq$!0[-#p'[O.$Ffy7$$\"0]7. " 0 "" {MPLTEXT 1 0 56 "P2:=plot(phi_a[1]/subs(x=0,phi_a[1] ),x=0..4,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "pl ots[display](P1,P2);" }}{PARA 13 "" 1 "" {GLPLOT2D 695 225 225 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"0mm;a)G\\a!#;$\"0 oUV[q,)**!#:7$$\"0Lek`o!>5F2$\"0&*>'o!42$**F27$$\"0n;z>)G_:F2$\"0$G$pR z%R)*F27$$\"0n;aQU!*3#F2$\"0F%4/k%*4(*F27$$\"0LeRZXKi#F2$\"05L+*Q/W&*F 27$$\"0m\"z>,_=JF2$\"0.-`\"H&zN*F27$$\"0+D\"G$[8j$F2$\"08iD^GetXF27$$\"0]7`H\"fT5!#9$\"0na=VNp7%F27$$\"0] Pf)[$H4\"Fjq$\"0v!)4W&)Gr$F27$$\"0Lek`1l9\"Fjq$\"0ei3XEnH$F27$$\"0$e*[ .-d>\"Fjq$\"0<)pOW5KHF27$$\"0nTg-m([7Fjq$\"09tIDI2c#F27$$\"0F27$$\"0n\"zWho.9Fj q$\"0cs-5_'G;F27$$\"0++D'>Ad9Fjq$\"0,H)\\Rvj8F27$$\"0+DcJ'f4:Fjq$\"02' GE.hL6F27$$\"0]7`>r-c\"Fjq$\"0&H\"R'Q!pP*F/7$$\"0+v$4q`;;Fjq$\"0&ycAIF (\\(F/7$$\"0LLeM%4n;Fjq$\"0T^)\\18egF/7$$\"0+]P4v5s\"Fjq$\"0Fr=ct?w%F/ 7$$\"0Fjq$\"0tK,Zycm\"F/7$$\"0L$3-=!y( >Fjq$\"0CL@-2HB\"F/7$$\"0]7G8O;.#Fjq$\"0u!H)z3\">))!#<7$$\"0nm;*\\[$3# Fjq$\"0!Q#*fwiLiF\\x7$$\"0nT&Qz]O@Fjq$\"0r[V:J+A%F\\x7$$\"0$ekG=4*=#Fj q$\"0By6#Hb(p#F\\x7$$\"0++D'4TPAFjq$\"0%*>q8ycf\"F\\x7$$\"0L3F9!z#H#Fj q$\"0*ePVFK1b!#=7$$\"0nm;%>KUBFjq$!0[-#p'[O.$Ffy7$$\"0]7.^7F2 $\"0;&*F27$$\"0nmm\"yYULF2$\"0dfu()y)Q\"*F27$$\"0LL$eF>(>%F2$\"0)Qc Bt:s')F27$$\"0mm;>K'*)\\F2$\"0QJ/:\")*p\")F27$$\"0++]Kd,\"eF2$\"0GtD/^ Hf(F27$$\"0mm;fX(emF2$\"0+$=Hw^]pF27$$\"0++]U7Y](F2$\"0^#pykP!G'F27$$ \"0LLLV!pu$)F2$\"0+zNf=ud&F27$$\"0mmmhb59*F2$\"0W1lpU.'\\F27$$\"0+++8! Q+5Fjq$\"0&e0)*H5#G%F27$$\"0+++&*3q3\"Fjq$\"0WuSwi=j$F27$$\"0+++(=\\q6 Fjq$\"0t'Gh,JXIF27$$\"0nm\"fBIY7Fjq$\"0UqL\"Qx`DF27$$\"0LLLO[kL\"Fjq$ \"0t=62ul-#F27$$\"0LLL&Q\"GT\"Fjq$\"0j?'>,\">j\"F27$$\"0++D2X;]\"Fjq$ \"0#*RszbVB\"F27$$\"0LLLvv-e\"Fjq$\"0r@xhh9P*F/7$$\"0++D2Ylm\"Fjq$\"0V cFjq$\"0VUyz]h#=F/7$$\"0nm;kD!)*>Fjq$\"0YJ!zOIP%*F\\x7 $$\"0nm\"f`@'3#Fjq$\"06\\2kNVT$F\\x7$$\"0++vw%)H;#Fjq$\"0/f)>Iv*H#Ffy7 $$\"0nm;$y*eC#Fjq$!0P[yk#*ee\"F\\x7$$\"0+++9b:L#Fjq$!0qU4^h/?#F\\x7$$ \"0++]5a`T#Fjq$!0VN;EQ=+#F\\x7$$\"0++D\"RV'\\#Fjq$!09X>+w@R\"F\\x7$$\" 0++]@fke#Fjq$!07Rp%)Ffy7$$\"0nm\"zM)>$GFjq$\"0VGnL=?\"F\\x7$$\"0LLL)G[kJFjq$\"0k7q!\\g0\"*Ffy7$$\"0++ D\"yh]KFjq$\"0avgyuv[&Ffy7$$\"0nmm)fdLLFjq$\"0M54yL\"y>Ffy7$$\"0nm;q7% =MFjq$!05^1.,/?\"Ffy7$$\"0LLe#pa-NFjq$!0N%o)\\$fTOFfy7$$\"0+++ad)zNFjq $!0yV7<%z=^Ffy7$$\"0LL$GUYoOFjq$!0yfe?oU(eFfy7$$\"0nmm5:xu$Fjq$!0&H'*3 '=%pdFfy7$$\"0++D28A$QFjq$!0AwpBMx*\\Ffy7$$\"0++vS)38RFjq$!0y?!\\k?5QF fy7$$\"\"%F)$!0%=G`4^wAFfy-F][l6&F_[lF(F($\"*++++\"!\")-%+AXESLABELSG6 %Q\"x6\"Q!6\"%(DEFAULTG-%%VIEWG6$;F(F^[mF^\\m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 192 "C heck the accuracy of the eigenenergies of the first and second excited states, and compare the graphs of the eigenfunctions as obtained by m atrix diagonalization versus numerical integration." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 11 "Exercise 3:" }}{PARA 0 "" 0 "" {TEXT -1 102 "Check the results obtained for different choic es of the box size that defines the free-particle basis." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 478 "It is pos sible to assess the accuracy of the matrix diagonalization without ref erring to the numerical calculation. One can calculate the expectation value of the Hamiltonian in the coordinate representation for the giv en eigenstate. This agrees with the eigenvalue obtained from the matri x diagonalization. Then, one can ask about the distribution of energie s in the state. The deviation from the average is calculated from the \+ matrix elements of the square of the Hamiltonian." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 209 "The functions to calcula te these are defined below. However, they work only when the truncatio n size of the matrix is relatively small. Otherwise too much memory is consumed in the evaluation of the integrals." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Havg:=psi->int(simplify(psi*(-1/2*diff(psi,x$2)+ V*psi)),x=-X..X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%HavgGR6#%$psiG 6\"6$%)operatorG%&arrowGF(-%$intG6$-%)simplifyG6#*&9$\"\"\",&-%%diffG6 $F3-%\"$G6$%\"xG\"\"##!\"\"F=*&%\"VGF4F3F4F4F4/F<;,$%\"XGF?FEF(F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "E1_avg:=Havg(phi_a[1]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'E1_avgG$\"-f2WX3n!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "H2me:=psi->int(simplify((-1/2*diff( psi,x$2)+V*psi)^2),x=-X..X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%H2m eGR6#%$psiG6\"6$%)operatorG%&arrowGF(-%$intG6$-%)simplifyG6#*$),&-%%di ffG6$9$-%\"$G6$%\"xG\"\"##!\"\"F=*&%\"VG\"\"\"F8FBFBF=FB/F<;,$%\"XGF?F FF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Edev:=sqrt(abs(E 1_avg^2-H2me(phi_a[1])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%EdevG$ \"0]=s0FAm#!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 283 "The deviation \+ is much larger than what we would have expected from our comparison wi th the numerical value. Nevertheless, the number is proportional to th e accuracy for the given state: when the matrix size is increased the \+ deviation from the average of the energy decreases for all " }{TEXT 280 1 "x" }{TEXT -1 49 ". The large deviation is likely to come from t he " }{TEXT 281 1 "x" }{TEXT -1 124 "-range where the approximate wave function is oscillating. This can be verified by restricting the calc ulation to a smaller " }{TEXT 282 1 "x" }{TEXT -1 61 "-range where the true ground-state eigenfunction is non-zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 152 "Another way to explore t he accuracy of the matrix diagonalization result is to graph the local energy in position space. For an eigenstate the function " }{TEXT 284 1 "f" }{TEXT -1 1 "(" }{TEXT 283 1 "x" }{TEXT -1 57 ") = (H*psi)/p si should be equal to the eigenvalue at all " }{TEXT 285 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:=psi->(-1/2*diff(psi,x$2))/psi+V;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%$psiG6\"6$%)operatorG%&arrowGF(,&*&-%%diff G6$9$-%\"$G6$%\"xG\"\"#\"\"\"F1!\"\"#F8F6%\"VGF7F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(f(phi_a[1]),x=0..3,view=[0..3, 0..2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 693 223 223 {PLOTDATA 2 "6%-%'CU RVESG6$7`r7$$\"\"!F)$\"0h'=GG&Q,)!#:7$$\"0+++DY\"Rl!#;$\"09_]9]&pzF,7$ $\"0+]PC#)GA\"F,$\"0oS%e1?gyF,7$$\"0++v$eui=F,$\"0x'))fd%Gm(F,7$$\"0++ D'3&o]#F,$\"0lyh1fS*\\F,$\"0ya< v3h\"fF,7$$\"0+](=$f%GcF,$\"0S*fb.<;bF,7$$\"0++]#y,\"G'F,$\"0O5B\">'49 &F,7$$\"0++Dr\"zboF,$\"0e:18;L'[F,7$$\"0++](4&G](F,$\"0;uI'f,PYF,7$$\" 0++]7nD:)F,$\"0r&*GF&)=`%F,7$$\"0++]-*oy()F,$\"0lWrW/ld%F,7$$\"0+]Ppns M*F,$\"0[q2+xdw%F,7$$\"0++DFOB+\"!#9$\"0[jW-!=0_F,7$$\"0+++R5'f5Ffp$\" 0:2&oWbC*Ffp7$$\"0+DJ'=_68Ffp$\"0usO1^%*3 \"Ffp7$$\"0+]P%y!eP\"Ffp$\"0@F34+gG\"Ffp7$$\"0+v=WU[V\"Ffp$\"09/F>yK[ \"Ffp7$$\"0+]7B>&)\\\"Ffp$\"0%RxT:$Qq\"Ffp7$$\"0+v$>:mk:Ffp$\"0C,tFfp7$$\"0+DcdQAi\"Ffp$\"0:J%\\.%p2#Ffp7$$\"0+]PPBWo\"Ffp$\"0G#H%GPF;# Ffp7$$\"0++]Nm'[Ffp$!/W#fw1Xb\"F_u7$$\"0+ DJ\")y,(>Ffp$!/aL>mxF_u7$$\"0v=n[k*z?Ffp$! 0z*4#))*e)[#F_u7$$\"0]P4')QY4#Ffp$!0Fs5!\\v8LF_u7$$\"0Dc^B8$4@Ffp$!06A E,Heb%F_u7$$\"0+v$4w)R7#Ffp$!0)Q[&)4F4mF_u7$$\"0i!*G44?8#Ffp$!0-*R`f62 %)F_u7$$\"0D1kdI+9#Ffp$!0q:qF:i6\"!#77$$\"01k\"=8/W@Ffp$!0%en*ehpJ\"Fg x7$$\"0(=#*f?0[@Ffp$!0Z&Ghed*e\"Fgx7$$\"0y+3Vd+:#Ffp$!0le?v#)fw\"Fgx7$ $\"0pz;!G1_@Ffp$!09w=I&o!)>Fgx7$$\"0feD&G%3g @Ffp$!0:_*ftBgOFgx7$$\"0Augl*3i@Ffp$!0&zy&Qu]d%Fgx7$$\"08`p-&4k@Ffp$!0 m29QMN0'Fgx7$$\"0.KyR+h;#Ffp$!0wxH,Kv%))Fgx7$$\"0%4rod5o@Ffp$!0#o5>/v6 ;!#67$$\"0&)*eR66q@Ffp$!0kwF\\eV=)Fd\\l7$$\"0vo/^;@<#Ffp$\"0)4EMMUdt[T=#Ffp$\"0\"*3R)fRiKFgx7 $$\"05@m5ah=#Ffp$\"0ak;\\dH(GFgx7$$\"0++vZf\")=#Ffp$\"0(R]7mosDFgx7$$ \"0%[@vj/!>#Ffp$\"07k!)*4(oM#Fgx7$$\"0pHHFL>>#Ffp$\"0;OzlT6;#Fgx7$$\"0 `W1#Ffp$\"0e>@IAd+#Fgx7$$\"0Qf$oqq&>#Ffp$\"0A@\">nxt=Fgx7$$\"02*yj 3[*>#Ffp$\"0zxc#y*=m\"Fgx7$$\"0v=#fYD.AFfp$\"0t^%)\\F$*\\\"Fgx7$$\"08y +D-3@#Ffp$\"0A#[^/cm7Fgx7$$\"0]P4%)\\$=AFfp$\"0W\"4QYG36Fgx7$$\"0DcE-X MB#Ffp$\"0P?b#ody!*F_u7$$\"0+vV?S&[AFfp$\"0n5_LRJ(yF_u7$$\"0+DJ$y4!G#F fp$\"0>]C5\">zkF_u7$$\"0+v=Yb;J#Ffp$\"0Y!*>B)ekdF_u7$$\"0](=7)3DM#Ffp$ \"0u'*))eXMO&F_u7$$\"0++D;iLP#Ffp$\"0)fGqKo>^F_u7$$\"0]P4wicS#Ffp$\"03 V/6vA'\\F_u7$$\"0+v$fL'zV#Ffp$\"0Nki+?F'[F_u7$$\"0+++*>=+DFfp$\"0)*4xt e`#Ffp$\"0D:iU;fg%F_u7$$\"0+]i_4Qc#Ffp$\"0gI'f9O`V F_u7$$\"0D\"G$p%ezDFfp$\"0Ue`V5=5%F_u7$$\"0]7.')f`f#Ffp$\"09cN8;!\\OF_ u7$$\"0vVt-N6h#Ffp$\"0jVv'>J%p#F_u7$$\"0+vV>5pi#Ffp$!/Ep'G(3#)>F_u7$$ \"019%HT`IEFfp$!0ji\\(*QY(=F_u7$$\"07`W1eTj#Ffp$!06o&*>G6&[F_u7$$\"0ls >.qfj#Ffp$!02qe#4^0uF_u7$$\"0=#\\**>yPEFfp$!0V;!oGab6Fgx7$$\"0r6q'RfRE Ffp$!0$)G/ORj%>Fgx7$$\"0DJX$fSTEFfp$!0ShY\"))*>/%Fgx7$$\"0x]?!z@VEFfp$ !0]#Rb)Gwd#Fd\\l7$$\"0Iq&p)H]k#Ffp$\"0L=W1b:w(Fgx7$$\"0$)*3P=%ok#Ffp$ \"0eP_9=Uv$Fgx7$$\"0P4Y!Ql[EFfp$\"0/LkM\\Qi#Fgx7$$\"0!*G@xl/l#Ffp$\"0g _a%G0\"4#Fgx7$$\"0V['RxF_EFfp$\"0STq?%H\"y\"Fgx7$$\"0(z;2(*3aEFfp$\"03 %4[?%*y:Fgx7$$\"0](ou;!fl#Ffp$\"0V*4E*=lV\"Fgx7$$\"0jlZa\\Jm#Ffp$\"0u! HEtyK6Fgx7$$\"0vV[T(RqEFfp$\"0dsHV,j&**F_u7$$\"0)=#\\GXwn#Ffp$\"0Ah][W p=*F_u7$$\"0++]:$*[o#Ffp$\"0Y$H2K/.()F_u7$$\"0]i!4p],FFfp$\"0yeG#om$4) F_u7$$\"0+DJm?\"=FFfp$\"0_#*zm0U!yF_u7$$\"0Dc,aFks#Ffp$\"0dA=PP*>xF_u7 $$\"0](=bF%3V)F_u7$$\"\"$F)$\"0Zlh#R%\\&*)F_u-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fg_m;F($\"\"#F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 451 "This graph should have a sobering effect for any enthusiasm that may have developed after obtaining a r easonably accurate eigenvalue, and an approximate eigenfunction that a ppeared to follow the numerically calculated one: The local energy cal culation is a highly sensitive quantity. In particular, we notice that the basis-state representation in the 'particle-in-a-box' basis has d ifficulties with obtaining the correct fall-off of the wavefunction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 444 "We sho uld not be deterred too much by this mixture of success and disappoint ment: we should be aware of the fact that eigenenergies are much easie r to obtain than accurate eigenfunctions, and proceed with cautiously. Matrix diagonalization is often the only tool available to solve the \+ Schroedinger equation. Usually we also have the possibility to improve matters by adjusting the basis to the problem at hand, and thereby re ducing the errors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 11 "Exercise 4:" }}{PARA 0 "" 0 "" {TEXT -1 218 "Graph the l ocal energy for matrix diagonalization solutions differing by matrix s ize, and observe how the better converged calculations approximate the eigenvalue locally. Repeat the calculation for a smaller value of " } {TEXT 287 1 "X" }{TEXT -1 7 " (with " }{TEXT 288 1 "X" }{TEXT -1 112 " larger than the point at which the numerically obtained wavefunction \+ goes to zero), and make your observations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "We carry out the calcul ation for the first excited state:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ev_s[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3<8%oza ,)4C!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "IC:=u(0)=0,D(u)( 0)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6$/-%\"uG6#\"\"!F*/--% \"DG6#F(F)\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E_t:=2. 392;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "SE:=-1/2*diff(u(x),x$2)+(V- E_t)*u(x)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sol:=dsolve(\{SE,IC \},u(x),numeric,output=listprocedure):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u_s:=subs(sol,u(x)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E_t G$\"%#R#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SEG/,&-%%diffG6$-% \"uG6#%\"xG-%\"$G6$F-\"\"##!\"\"F1*&,&*$)F-\"\"%\"\"\"F9$\"%#R#!\"$F3F 9F*F9F9\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "P1:=plot('u _s(x)',x=0..2.5): plots[display](P1,view=[0..2.5,-1..1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 693 187 187 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$$\"\"!F) F(7$$\"0LL3FWYs#!#;$\"0J=()zJIs#F-7$$\"0mm;a)G\\aF-$\"0VQ9k&ROaF-7$$\" 0)\\7`p)*>yF-$\"0BWca8>y(F-7$$\"0Lek`o!>5!#:$\"0O'[@8l55F=7$$\"0](=n$y cG\"F=$\"0OnXp+)o7F=7$$\"0n;z>)G_:F=$\"0Y,_=JF=$\"0&*>t$oR#)GF=7$$\"0+D\"G$[8j$F=$\"0x'>pZxhKF=7$$\"0 m\"z%*frhTF=$\"0p9ZWy7h$F=7$$\"0+Dcw#Q!p%F=$\"0YG`3w>\"RF=7$$\"0LL3_\" =M_F=$\"0>lS0hy;%F=7$$\"0m;/wfJr&F=$\"0+3'eJ3YVF=7$$\"0++D\"eP_iF=$\"0 #*=f\"*>A\\%F=7$$\"0++](34BlF=$\"075,9qNa%F=7$$\"0++v$f!Qz'F=$\"0f(40E ?!e%F=7$$\"0++D1!paqF=$\"0[_vF=$\"0[w!z\"fbg%F=7$$\"0m\"zW(*Q*y(F=$\"0eeTT^%F=7$$\"0LLLe'3I) )F=$\"0\\u`g6kS%F=7$$\"0*\\7.T(pnG9QF=7$$\"0]Pf)[$H4\"Fjt$\"0<\" \\'oxUc$F=7$$\"0Lek`1l9\"Fjt$\"0Q1Ia)=%G$F=7$$\"0$e*[.-d>\"Fjt$\"0(z,@ *od,$F=7$$\"0nTg-m([7Fjt$\"0iV'3!y1s#F=7$$\"0Ad9Fjt$\"0#)3ONQ&>;F=7$$\"0+DcJ'f4:Fjt$\"0rKO([A!Q\"F=7$$ \"0]7`>r-c\"Fjt$\"0sV^!pxo6F=7$$\"0+v$4q`;;Fjt$\"03;Tjn\")e*F-7$$\"0LL eM%4n;Fjt$\"0@&)yY8%F-7$$\"0$ek.Nyt=Fjt$\" 0fnKhIVI$F-7$$\"0]i:bzj#>Fjt$\"07'yCQ&Qh#F-7$$\"0L$3-=!y(>Fjt$\"0e*H!G e\")4#F-7$$\"0]7G8O;.#Fjt$\"0si+m'*=r\"F-7$$\"0nm;*\\[$3#Fjt$\"0q`hF,z Z\"F-7$$\"0nT&Qz]O@Fjt$\"0U>&R>ju8F-7$$\"0$ekG=4*=#Fjt$\"0i'e8iA99F-7$ $\"0++D'4TPAFjt$\"04iikVKf\"F-7$$\"0L3F9!z#H#Fjt$\"0?0')f$R6?F-7$$\"0n m;%>KUBFjt$\"01*fcW-]EF-7$$\"0]7. 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The plot shows size of the tunneling region, as well as the fact that the region where the numerical eigenfunction touches down to the " } {TEXT 266 1 "x" }{TEXT -1 293 "-axis (it diverges for large x, due to \+ the finite accuracy of the trial eigenenergy, and due to numerical pro blems) is well inside the classically forbidden region for the given s tate. For higher-lying states one needs to go to larger x-values for t he 'touchdown' point. Eventually, for large " }{TEXT 267 1 "n" }{TEXT -1 86 " the basis set for the matrix diagonalization will be inappropr iate due to the finite " }{TEXT 268 1 "X" }{TEXT -1 7 "-value." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 714 " It is important to explore the accuracy of states higher than the grou nd states of the even- and odd-parity sectors. These states are harder to calculate, since one needs to find the locations of nodes that are not pre-determined. The eigenstates calculated from the matrix diagon alization method are all mutually orthogonal. Given their approximate \+ nature, however, one cannot claim that the second excited state is ort hogonal to the exact ground state. Experience shows that the accuracy \+ of the higher-lying eigenvalues (and their eigenfunctions) is substant ially less than what was found for the two lowest states (for the same hamiltonian matrix). One can try to beat the problem by increasing th e matrix size." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We demonstrate the problem on the example of the first ex citation in the even-parity sector:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ev_s[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3D7E8t/ aA`!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "IC:=u(0)=1,D(u)(0 )=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6$/-%\"uG6#\"\"!\"\"\"/- -%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E_t:=4.69 7;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "SE:=-1/2*diff(u(x),x$2)+(V-E_ t)*u(x)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sol:=dsolve(\{SE,IC\} ,u(x),numeric,output=listprocedure):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u_s:=subs(sol,u(x)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E_tG$ \"%(p%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SEG/,&-%%diffG6$-%\"u G6#%\"xG-%\"$G6$F-\"\"##!\"\"F1*&,&*$)F-\"\"%\"\"\"F9$\"%(p%!\"$F3F9F* F9F9\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "P1:=plot('u_s( x)',x=0..3): plots[display](P1,view=[0..3,-1.1..1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 693 187 187 {PLOTDATA 2 "6%-%'CURVESG6$7\\o7$$\"\"!F)$\" \"\"F)7$$\"0++Dc'yM;!#;$\"0E>iw\\u)**!#:7$$\"0++]7t&pKF/$\"0(=Jn0$)\\* *F27$$\"0++vofV!\\F/$\"09Zg$oB())*F27$$\"0+++DY\"RlF/$\"0bP>rD)*z*F27$ $\"0+]PM%)RQ*F/$\"0yhL)*H#*e*F27$$\"0+]PC#)GA\"F2$\"0M,o1ydI*F27$$\"0+ +v$eui=F2$\"0jUq9TST)F27$$\"0++D'3&o]#F2$\"0sA`2(y!>(F27$$\"0+](oX*y9$ F2$\"0;@VC\\hp&F27$$\"0+]P9CAu$F2$\"0k>&[[j8TF27$$\"0+]P*zhdVF2$\"0AgU OtGL#F27$$\"0+]P>fS*\\F2$\"0&[j:_zbSF/7$$\"0+](=$f%GcF2$!0JB/j)zI:F27$ $\"0++]#y,\"G'F2$!0.oq()QJY$F27$$\"0++Dr\"zboF2$!0Z#fT))*p0&F27$$\"0++ ](4&G](F2$!0\"zgaT'4n'F27$$\"0++]7nD:)F2$!04=v'=UY!)F27$$\"0++]-*oy()F 2$!0)pYf!)***4*F27$$\"0+]PpnsM*F2$!0OTXKBX!)*F27$$\"0+v$4_J&o*F2$!0Fok To0,\"!#97$$\"0++DFOB+\"F\\r$!0i[lTy=.\"F\\r7$$\"0+]7Lt4.\"F\\r$!0_041 EJ/\"F\\r7$$\"0+++R5'f5F\\r$!0?-'*Gv#[5F\\r7$$\"0](=(4AH4\"F\\r$!0,O)o J&o/\"F\\r7$$\"0+vV!QBE6F\\r$!0*R3+1%y.\"F\\r7$$\"0](o4.sb6F\\r$!01`aH ;R-\"F\\r7$$\"0++]\"o?&=\"F\\r$!0NEQ!*3[+\"F\\r7$$\"0+vVb4*\\7F\\r$!0% RK.Rbo%*F27$$\"0+DJ'=_68F\\r$!09)>oD%*\\()F27$$\"0+]P%y!eP\"F\\r$!0$p- =0yxyF27$$\"0+v=WU[V\"F\\r$!0ke,R(\\6qF27$$\"0+]7B>&)\\\"F\\r$!0I>le?E 0'F27$$\"0+v$>:mk:F\\r$!0O$G\"3*Qu]F27$$\"0+DcdQAi\"F\\r$!0'>$>gnyE%F2 7$$\"0+]PPBWo\"F\\r$!0N\"3=L\\nMF27$$\"0++]Nm'[vT#[=;F27$$\"0+]7T W)R>F\\r$!0'Qr`\"p#p6F27$$\"0++]@80+#F\\r$!0M$)4V2%=&)F/7$$\"0++D6!Hl? 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M8^'zMYF\\r7$Feo$!0WyQ9yjV$F\\r7$Fbs$!0'*y'* )[:)3$F\\r7$F\\t$!0mRGAvPs#F\\r7$Fat$!0md(z-IcAF\\r7$Fft$!0aFnx\"HQQ6\\Af(Fejm7$Fhw$\"/Oy `99j%*Fejm7$F]x$\"0FV-#H%>8\"Fejm7$Fbx$\"00K\"\\No\\8Fejm7$Fgx$\"0nf,& Q]l:Fejm7$F\\y$\"0.FFejm7$Faz$\"08f$pd,jIFejm7$Ffz$\"0Q2 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 96 "Solution of \+ the numerical problem by bisection for the general case of non-symmetr ic potentials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We start the integration from " }{TEXT 272 1 "x" }{TEXT -1 3 "= -" }{TEXT 271 1 "s" }{TEXT -1 5 " and " }{TEXT 270 2 "x " } {TEXT -1 2 "= " }{TEXT 269 1 "s" }{TEXT -1 108 ", and try to match the m at the origin, by requesting that u'(0)/u(0) match from the left and from the right." }}{PARA 0 "" 0 "" {TEXT -1 418 "The boundary conditi on at these two points distinguishes even- and odd-parity states. For \+ even-parity states the derivatives of the wavefunction are opposite to each other, for odd-parity states they are identical. The choices det ermine the normalization of the propagated solution. Given that we mat ch u'(0)/u(0) this should not matter at all. Let us illustrate what h appens as we integrate from the outside inwards." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s:=3.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" sG$\"\"$\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eta:=3*10^ (-4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "IC1:=u(s)=0,D(u)(s )=eta: IC2:=u(-s)=0,D(u)(-s)=-eta:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E_t:=4. 697;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "SE:=-1/2*diff(u(x),x$2)+(V- E_t)*u(x)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sol1:=dsolve(\{SE,I C1\},u(x),numeric,output=listprocedure): u_1:=subs(sol1,u(x)):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sol2:=dsolve(\{SE,IC2\},u(x),numeri c,output=listprocedure): u_2:=subs(sol2,u(x)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E_tG$\"%(p%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#SEG/,&-%%diffG6$-%\"uG6#%\"xG-%\"$G6$F-\"\"##!\"\"F1*&,&*$)F-\"\"%\" \"\"F9$\"%(p%!\"$F3F9F*F9F9\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 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{MPLTEXT 1 0 55 "S E:=-1/2*diff(u(x),x$2)+(V-E_t)*u(x)=0; eta:=3*10^(-4):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "IC1:=u(s)=0,D(u)(s)=eta: IC2:=u(-s)=0,D(u)(-s) =-eta:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sol1:=dsolve(\{SE,IC1\},u (x),numeric,output=listprocedure): u_1:=subs(sol1,u(x)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sol2:=dsolve(\{SE,IC2\},u(x),numeric,output=l istprocedure): u_2:=subs(sol2,u(x)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "up_1:=subs(sol1,diff(u(x),x)): up_2:=subs(sol2,diff(u(x),x)):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "up_1(0)/u_1(0)-up_2(0)/u_2(0); end :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "The bisection algorithm is \+ written with two procedures: a basic bisection step, and a driver whic h reports on progress:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 " Bis1:=proc(x1,x2,x3,f1,f2,f3);\nif evalf(f1*f3) < 0 then\nRETURN([x1,x 3,f1,f3]);\nelse\nRETURN([x3,x2,f3,f2]); fi;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "Bisect:=proc(f,a,b) local res,x1,x2,x3,f1, f2,f3,i;x1:=a: x2:=b: f1:=evalf(f(x1)): f2:=evalf(f(x2)):\nif evalf(f1 *f2)>0 then RETURN(\"No bracketed root\",f1,f2);\nelse\nx3:=0.5*(x1+x2 ); f3:=evalf(f(x3)); fi;\nfor i from 1 to 50 do:\nres:=Bis1(x1,x2,x3,f 1,f2,f3);\nx1:=res[1]; f1:=res[3]; x2:=res[2]; f2:=res[4]; x3:=0.5*(x1 +x2);\nif abs(x1-x2) < 10^(-2) then print(\"Reached level 2\",x3); fi; if abs(x1-x2) < 10^(-7) then RETURN(x3) fi;\nf3:=evalf(f(x3)); od:\nR ETURN(\"Loop exhausted\", x3); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "s:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Bisect(Match,4.65,4.75); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\")](op%!\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\"*]7`p%!\") " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\"+]P4'p%!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\",]P%['p%!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\"-](ozmp%!#6 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\".]PMxnp%!#7 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\"/](=o`z'p%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~le vel~26\"$\"0JyeT&z'p%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0JyeT&z 'p%!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Let us pick a non-symme tric potential and calculate the eigenvalues:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "V:=x^4-2*x^ 3-x^2/2+2/2*x+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,,*$)%\"xG\" \"%\"\"\"F**&\"\"#F*)F(\"\"$F*!\"\"*&#F*F,F**$)F(F,F*F*F/F(F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "P0:=plot(V,x=-5..5,view=[-2. .3,-1..5],color=green): plots[display](P0);" }}{PARA 13 "" 1 "" {GLPLOT2D 713 201 201 {PLOTDATA 2 "6%-%'CURVESG6$7X7$$!\"&\"\"!$\"0+++ ++]e)!#77$$!0mm\"HU,\"*[!#9$\"024#z!pT!zF-7$$!0LL$e%G?y%F1$\"0BPgz!Hks F-7$$!0+v=_+so%F1$\"0)Q<8IcRnF-7$$!0n;aesBf%F1$\"0&4KE?^ViF-7$$!0+DJlG 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"6$Q0Reached~level~26\"$\",D\"yD*)*)!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\"-v=#*p\"**)!# 7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\".v=#*>H**) !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\"/vVt-`$* *)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\"0Dcw4D K**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\"0il( 4D2$**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$\"0 $4r.)[J**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26\"$ \"0e$o]p=$**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level~26 \"$\"0D(>xy;$**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~level ~26\"$\"0TSRTxJ**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached~lev el~26\"$\"0*>J#=#=$**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Reached ~level~26\"$\"0z(\\mX=$**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q0Rea ched~level~26\"$\"0!\\SuL=$**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q 0Reached~level~26\"$\"0N^/(R=$**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$Q0Reached~level~26\"$\"08GCn$=$**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"08GCn$=$**)!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "E 1:=%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G$\"08GCn$=$**)!#:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "To graph the solution we use the \+ obtained best eigenvalue, and integrate the Schroedinger equation once more:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eta:=1*10^(-6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "IC1:=u(s)=0,D(u)(s)=eta: \+ IC2:=u(-s)=0,D(u)(-s)=eta:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "E_t:=E1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "SE:=-1/2*diff(u(x),x $2)+(V-E_t)*u(x)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sol1:=dsolve (\{SE,IC1\},u(x),numeric,output=listprocedure): u_1:=subs(sol1,u(x)): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "sol2:=dsolve(\{SE,IC2\},u(x),nu meric,output=listprocedure): u_2:=subs(sol2,u(x)):" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$E_tG$\"08GCn$=$**)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SEG/,&-%%diffG6$-%\"uG6#%\"xG-%\"$G6$F-\"\"##!\"\"F1 *&,,*$)F-\"\"%\"\"\"F9*&F1F9)F-\"\"$F9F3*&#F9F1F9*$)F-F1F9F9F3F-F9$\"0 (=dFj\"o+\"!#:F9F9F*F9F9\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "P1:=plot('u_1(x)',x=-s..s,color=red): P2:=plot('u_2(x)',x=-s..s, color=blue): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "P00:=plot( E1,x=-3..3,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plots[display](P00,P0,P1,P2,view=[-3..3,-1..6]);" }}{PARA 13 "" 1 "" {GLPLOT2D 693 187 187 {PLOTDATA 2 "6(-%'CURVESG6$7S7$$!\"$\"\"!$\"08GC n$=$**)!#:7$$!0++]2<#pG!#9F+7$$!0+]7bBav#F1F+7$$!0++D$3XFEF1F+7$$!0++v #)H')\\#F1F+7$$!0+]i3@/P#F1F+7$$!0+]7iUC\"F1F+7$$!0+]7YY08\"F1F+7$$!/ ++]XF`**F1F+7$$!/+++Az2))F1F+7$$!/+]7RKvuF1F+7$$!/+++P'eH'F1F+7$$!/+]7 *3=+&F1F+7$$!/+]PFcpPF1F+7$$!/++DJ%Q[#F1F+7$$!/+]i6:.8F1F+7$$!-++v`hHF 1F+7$$\"/+](QIKH\"F1F+7$$\"/+]7:xWCF1F+7$$\"/++vuY)o$F1F+7$$\"/+++rKt 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Soluti on 1 (which is integrated left-to-right), is small, and cannot be seen on this scale (it is covered by the " }{TEXT 289 1 "x" }{TEXT -1 60 " -axis). Both solutions are normalized through the choice of " }{TEXT 19 3 "eta" }{TEXT -1 135 ". It is interesting to observe how the wavef unction reaches through the classically forbidden regime to extend ove r the second minimum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 11 "Exercise 5:" }}{PARA 0 "" 0 "" {TEXT -1 115 "Calculat e some of the excited eigenstates for this potential by supplying othe r bracketing values to the procedure " }{TEXT 19 6 "Bisect" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 11 " Exercise 6:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "Modify the potential function such that there be one broad, and one sharp minimu m with approximately equal depth by choosing different coeffients for \+ the monomials in " }{TEXT 292 1 "x" }{TEXT -1 162 ". Calculate the gro und state and observe its shape. When does it become double-humped, an d when does the second hump (over the narrower potential well) disappe ar?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 23 "Matrix diagonalization:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 172 "We can get an idea about the location of the low-lying eigenvalues by carrying out the matrix diagonalization. We simply copy the lines from the beginning of the worksheet." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "HM:=Matrix(N):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "We make use of the symmetry of the hamil tonian matrix: it allows to save almost a factor of 2 in computation t ime." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "for i from 1 to N \+ do: for j from 1 to i do: HM[i,j]:=Vpot(uB(i),uB(j)); if i=j then HM[i ,j]:=HM[i,j]+ Tkin(uB(i)); fi: od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for i from 1 to N do: for j from i+1 to N do: HM[i,j] :=HM[j,i]: od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "print (SubMatrix(HM,1..4,1..4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABL EG6$\")/vU9-%'MATRIXG6#7&7&,&*&,(*$)%#PiG\"\"%\"\"\"\"&V@\"*&\"'+[CF3) F1\"\"#F3!\"\"\"(gXZ\"F3F3*$F0F3F9#F3\"#:*&#F3\"$7&F3F7F3F3,$*&,&*$F7F 3\"%\\6\"&S-\"F9F3*$F0F3F9#!$c#\"#F,$*&,&!$G\"F3*&\"# " 0 "" {MPLTEXT 1 0 19 "HMf:=map(evalf,HM):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evals:=Eigenvalues(HMf, output='list'):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ev_s:=sort(map(Re,evals));" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%%ev_sG76$\"3!RlTH\"f;P#*!#=$\"3)34hE9=r)>!#<$\"3@ty lV)3z#QF+$\"3yrN@!4\\+'oF+$\"3sy1f+8>78!#;$\"3qM(>%>U!Hk#F2$\"3X@>\"f/ vDa%F2$\"3)3Yw>`A)f#)F2$\"38\\eQ05qK7!#:$\"3mFZ#)pj#*G?F;$\"3w)*y8;Q*z v#F;$\"3]:^a\"39AA%F;$\"3KUO\\-gP&R&F;$\"3$Rf2--3e#yF;$\"3S0@XGYD&e*F; $\"3\")3'pIPBBL\"!#9$\"3e(zZHanVe\"FJ$\"3M(3neX[N7#FJ$\"3'[!R0Ev%eZ#FJ $\"3['ocLkvnp$FJ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "VE:=Eig envectors(HMf,output='list'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Vp:=[seq([Re(VE[i][1]),VE[i][2],map(Re,VE[i][3])],i=1..nops(VE)) ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "VEs:=sort(Vp,Min):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 4 do:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "psi0:=add(VEs[i][3][j]*uB(j),j=1..N ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "No:=1/sqrt(int(expand(psi0^2) ,x=-X..X));" }}{PARA 0 "> 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}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "We observe that the matrix diagonalizati on gets the right idea about the ground-state eigenfunction (the eigen value is off by nearly 3 %)." }}{PARA 0 "" 0 "" {TEXT -1 83 "The first excited state has a bigger hump over the shallower part of the potent ial." }}{PARA 0 "" 0 "" {TEXT -1 180 "The higher states are not unlike the quartic harmonic oscillator eigenstates, but shifted to the right . For higher eigenenergies the particles explore mostly the steep rise in the " }{TEXT 19 3 "x^4" }{TEXT -1 23 "-part of the potential." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 11 "Exercise 7:" }}{PARA 0 "" 0 "" {TEXT -1 181 "Choose a potential with a deep an d narrow potential well, and make a detailed comparison between the gr ound state as obtained by matrix-diagonalization, and by the numerical method." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 46 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 14974976 14427504 }{RTABLE M6R0 I5RTABLE_SAVE/14974976X,%)anythingG6"6"][[[[[p1"%"%,&*&,(*$%#PiG""%""""$?"F-*$F +""#!#?F-F+!"%#"%'4%""&F/#F-"$7&""!,$*&,&!#:F-F/F0F-F+F2!%WhF8F8,&*&,("#:F-F/!# 5F*F0F-F+F2#"%[?F5F/#F-"$G"F8,$*&,&F1F-F/""$F-F+F2#!'s58"#FF9F8,&*&,(F/!#g"#SF- F*FMF-F+F2#F4"$N"F/#""*F7F8F8FGF8,&*&,(F/!#SF*"#KFAF-F-F+F2#FFF5F/#F-FenF& } {RTABLE M6R0 I5RTABLE_SAVE/14427504X,%)anythingG6"6"][[[[[p1"%"%,&*&,(*$%#PiG""%"&V@"*$F+""# !'+[C"(gXZ""""F2F+!"%#F2"#:F.#F2"$7&,$*&,&F."%\6!&S-"F2F2F+F3#!$c#"#F,$*&,&!$G" F2F."#