{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 }{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 "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 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 Outpu t" -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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 39 "Taylor expansion of an op erator product" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "We check the order of the split-operator time propagatio n technique. First, we analyze a popular version of the time splitting technique (used in Ehrenfest.mws). Then, we show why the simple opera tor product " }{TEXT 19 44 "exp(-I*(T+V)*dt) = exp(-I*T*dt) exp(-I*V*d t)" }{TEXT -1 16 " results in an " }{TEXT 19 7 "O(dt^2)" }{TEXT -1 10 " error if " }{TEXT 19 1 "T" }{TEXT -1 5 " and " }{TEXT 19 1 "V" } {TEXT -1 37 " do not commute. Then, we analyze an " }{TEXT 19 7 "O(dt^ 4)" }{TEXT -1 73 " accurate scheme which involves a large amount of al gebraic computations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "This is, of course, related to the Campbell-Baker-Hau sdorff formula, which reads ([A,B] : = A B - B A):" }}{PARA 0 "" 0 "" {TEXT -1 88 "exp(A) exp(B) = exp(O), where O = A + B + 1/2 [A,B] + 1/1 2 ([[A,B],B] + [A,[A,B]]) + ..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 110 "This worksheet can be used in maple6 and maple7, but it gobbles lots of memory there. It works fine in maple8. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "unassign(`&*`); \ndefine(`&*`,multi linear,zero=0,identity=1,flat);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A&*B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#&*G6$%\"AG %\"BG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a&*A;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%#&*G6$%\"aG%\"AG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "gamma &* A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%&ga mmaG\"\"\"%\"AGF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "If we need a variable to be treated as a constant, we add it to the list:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "constants:=constants,dt;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*constantsG6*%&falseG%&gammaG%)infin ityG%%trueG%(CatalanG%%FAILG%#PiG%#dtG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "&*(A,dt,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#dt G\"\"\"-%#&*G6%%\"AGF%%\"BGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#dtG\"\"\"-%# &*G6%%\"AGF%%\"BGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "tayl or(exp(-I*A&*B*dt),dt=0,3);" }}{PARA 8 "" 1 "" {TEXT -1 54 "Error, (in series/fracpower) unable to compute series\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Setting " }{TEXT 19 2 "dt" }{TEXT -1 112 " to a constan t is problematic, can't Taylor expand about it. So the trick is to Tay lor expand about a variable (" }{TEXT 19 1 "h" }{TEXT -1 228 " in our \+ case), and later to make substitutions, so that one may make use of th e constant property. The above Taylor expansion attempt failed, becaus e we can't expand about a variable that has just been declared to be a constant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "taylor(exp(-I *A&*B*h),h=0,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"hG\"\"\"\"\"! *&^#!\"\"F%-%#&*G6$%\"AG%\"BGF%F%,$*&#F%\"\"#F%*$)F*F2F%F%F)F2-%\"OG6# F%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(h=dt,%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*(^#!\"\"F$-%#&*G6$%\"AG%\" BGF$%#dtGF$F$*&#F$\"\"#F$*&)F(F0F$)F-F0F$F$F'-%\"OG6#*$)F-\"\"$F$F$" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "No:=5; #order to which we \+ expand" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NoG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Ut:=convert(taylor(exp(-I*(A+B)*h), h=0,No),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#UtG,,\"\"\"F&* (^#!\"\"F&,&%\"AGF&%\"BGF&F&%\"hGF&F&*(\"\"#F)F*F/F-F/F)*(^##F&\"\"'F& )F*\"\"$F&)F-F5F&F&*(\"#CF)F*\"\"%F-F9F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "We need to do something here: if we leave it as is, the p owers of " }{TEXT 19 3 "A,B" }{TEXT -1 46 " get evaluated assuming a c ommutative product." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "Ut:= subs((A+B)^2=(A+B)&*(A+B),(A+B)^3=(A+B)&*(A+B)&*(A+B),(A+B)^4=(A+B)&*( A+B)&*(A+B)&*(A+B),Ut);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#UtG,,\" \"\"F&*(^#!\"\"F&,&%\"AGF&%\"BGF&F&%\"hGF&F&*&#F&\"\"#F&*&,*-%#&*G6$F+ F+F&-F46$F+F,F&-F46$F,F+F&-F46$F,F,F&F&)F-F0F&F&F)*(^##F&\"\"'F&,2-F46 %F+F+F+F&-F46%F+F+F,F&-F46%F+F,F+F&-F46%F+F,F,F&-F46%F,F+F+F&-F46%F,F+ F,F&-F46%F,F,F+F&-F46%F,F,F,F&F&)F-\"\"$F&F&*&#F&\"#CF&*&,B-F46&F+F+F+ F+F&-F46&F+F+F+F,F&-F46&F+F+F,F+F&-F46&F+F+F,F,F&-F46&F+F,F+F+F&-F46&F +F,F+F,F&-F46&F+F,F,F+F&-F46&F+F,F,F,F&-F46&F,F+F+F+F&-F46&F,F+F+F,F&- F46&F,F+F,F+F&-F46&F,F+F,F,F&-F46&F,F,F+F+F&-F46&F,F,F+F,F&-F46&F,F,F, F+F&-F46&F,F,F,F,F&F&)F-\"\"%F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "UtA:=convert(taylor(exp(-I*(A)*h/2),h=0,No),polynom); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$UtAG,,\"\"\"F&*(^##!\"\"\"\"#F& %\"AGF&%\"hGF&F&*(\"\")F*F,F+F-F+F**(^##F&\"#[F&)F,\"\"$F&)F-F5F&F&*( \"$%QF*F,\"\"%F-F9F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "UtB :=convert(taylor(exp(-I*(B)*h),h=0,No),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$UtBG,,\"\"\"F&*(^#!\"\"F&%\"BGF&%\"hGF&F&*(\"\"#F)F* F-F+F-F)*(^##F&\"\"'F&)F*\"\"$F&)F+F3F&F&*(\"#CF)F*\"\"%F+F7F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Uspo:=simplify(subs(h=dt,UtA &* UtB &* UtA));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%UspoG,^z\"\"\" F&*&#F&\"#'*F&*&)%#dtG\"\"%F&-%#&*G6$*$)%\"AG\"\"$F&F3F&F&F&*&#F&\"#[F &*&F+F&-F/6$%\"BGF1F&F&F&*&#F&\"\"#F&*&)F,F>F&-F/6%F3F;F&F&F&!\"\"*&#F &\"$)GF&*&)F,\"\"'F&-F/6$*$)F;F4F&F1F&F&FC*&#F&\"%/BF&*&FHF&-F/6$F1F1F &F&FC*&#F&\"#7F&*&F+F&-F/6$FLF3F&F&F&*&#F&\"%3YF&*&)F,\"\")F&-F/6%F3FL *$)F3F-F&F&F&F&*&#F&F)F&*&FHF&-F/6%F3FL*$)F3F>F&F&F&FC*&FUF&*&F+F&-F/6 %F3FLF&F&F&F&*&#F&\"$%QF&*&FHF&-F/6%FcoF;F1F&F&FC*&#F&\"#;F&*&F+F&-F/6 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{XPPMATH 20 "6$\" \"!F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "C2:=coeff(Udif,dt, 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2G,.*&#\"\"\"\"\"#F(-%#&*G6 %%\"AG%\"BGF(F(F(*&#F(F)F(-F+6$F-F.F(!\"\"*&#F(F)F(-F+6$F.F.F(F3*&\"\" %F3F-F)F(*&F)F3F.F)F(*&#F(F9F(-F+6$F-F-F(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "C3:=coeff(Udif,dt,3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#C3G,B*&^##\"\"\"\"\"'F)-%#&*G6%%\"AGF.%\"BGF)F)*&F'F )-F,6%F.F.F.F)F)*&^##!\"\"\"#;F)-F,6$*$)F.\"\"#F)F.F)F)*&^##F6\"\"%F)- F,6$*$)F/FF)-F,6%F.FCF)F)F)*&F'F)-F,6%F/F.F.F)F)*&F'F)-F, 6%F.F/F/F)F)*&^##F6\"#7F)-F,6%F.F/F.F)F)*&^##F6F*F))F/\"\"$F)F)*&^##F6 \"#CF))F.FXF)F)*&F'F)-F,6%F/F/F/F)F)*&F'F)-F,6%F/F.F/F)F)*&F'F)-F,6%F/ F/F.F)F)*&^##F6\"\")F)-F,6%F:F/F)F)F)*&F4F)-F,6$F.F:F)F)*&FboF)-F,6$F/ F:F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "C4:=coeff(Udif,dt ,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#C4G,hn*&#\"\"\"\"#'*F(-%#&* G6$*$)%\"AG\"\"$F(F/F(!\"\"*&#F(\"#[F(-F+6$%\"BGF-F(F1*&#F(\"#7F(-F+6$ *$)F7F0F(F/F(F1*&#F(F:F(-F+6%F/F=F(F(F1*&#F(\"#;F(-F+6%*$)F/\"\"#F(F7F /F(F1*&#F(FEF(-F+6%FH*$)F7FJF(F(F(F1*&#F(FEF(-F+6%F/F7FHF(F1*&#F(\"\") F(-F+6%F/FOF/F(F1*&#F(F4F(-F+6%F-F7F(F(F1*&#F(\"#CF(-F+6&F/F/F/F/F(F(* &FinF(-F+6&F/F/F/F7F(F(*&FinF(-F+6&F/F/F7F/F(F(*&FinF(-F+6&F/F/F7F7F(F (*&FinF(-F+6&F/F7F/F/F(F(*&FinF(-F+6&F/F7F/F7F(F(*&FinF(-F+6&F/F7F7F/F (F(*&FinF(-F+6&F/F7F7F7F(F(*&FinF(-F+6&F7F/F/F/F(F(*&FinF(-F+6&F7F/F/F 7F(F(*&FinF(-F+6&F7F/F7F/F(F(*&FinF(-F+6&F7F/F7F7F(F(*&FinF(-F+6&F7F7F /F/F(F(*&FinF(-F+6&F7F7F/F7F(F(*&FinF(-F+6&F7F7F7F/F(F(*&FinF(-F+6&F7F 7F7F7F(F(*&\"$#>F1F/\"\"%F1*&FjnF1F7F\\rF1*&#F(\"#kF(-F+6$FHFHF(F1*&#F (FEF(-F+6$FOFHF(F1*&#F(F)F(-F+6$F/F-F(F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(C2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*& #\"\"\"\"\"#F&-%#&*G6%%\"AG%\"BGF&F&F&*&#F&F'F&-F)6$F+F,F&!\"\"*&#F&F' F&-F)6$F,F,F&F1*&\"\"%F1F+F'F&*&F'F1F,F'F&*&#F&F7F&-F)6$F+F+F&F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "definemore(`&*`,\n&*(1,A::s ymbol,B::symbol)=A&*B,&*(A::symbol,1,B::symbol)=A&*B,&*(A::symbol,B::s ymbol,1)=A&*B);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simpli fy(C2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"#F&-%#&*G6$% \"BGF+F&!\"\"*&\"\"%F,%\"AGF'F&*&F'F,F+F'F&*&#F&F.F&-F)6$F/F/F&F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "definemore(`&*`,&*(B::symbol ,B::symbol) = B^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "This one w as tricky, some other obvious versions didn't work (putting in the " } {TEXT 19 8 "::symbol" }{TEXT -1 16 " was essential)!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(A &* A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"AG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(C2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(C3);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,B*&^##\"\"\"\"\"'F'-%#&*G6%%\"AGF,%\" BGF'F'*&F%F'-F*6%F,F,F,F'F'*&^##!\"\"\"#;F'-F*6$*$)F,\"\"#F'F,F'F'*&^# #F4\"\"%F'-F*6$*$)F-F:F'F,F'F'*&F " 0 "" {MPLTEXT 1 0 14 "whattype(B^2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"^G" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 130 "definemore(`&*`,\n&*(1,A::algebraic,B::algebraic)= A&*B,&*(A::algebraic,1,B::algebraic)=A&*B,&*(A::algebraic,B::algebraic ,1)=A&*B);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(C3 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,B*&^##\"\"\"\"\"'F'-%#&*G6%%\"A GF,%\"BGF'F'*&^##!\"\"\"\")F'-F*6$*$)F,\"\"#F'F-F'F'*&F%F'-F*6%F,F,F,F 'F'*&^##F1\"#;F'-F*6$F5F,F'F'*&^##F1\"\"%F'-F*6$*$)F-F7F'F,F'F'*&F%F'- F*6%F-F,F,F'F'*&F%F'-F*6%F,F-F-F'F'*&^##F1\"#7F'-F*6%F,F-F,F'F'*&^##F1 F(F')F-\"\"$F'F'*&^##F1\"#CF')F,FYF'F'*&F%F'-F*6%F-F-F-F'F'*&F%F'-F*6% F-F,F-F'F'*&F%F'-F*6%F-F-F,F'F'*&F " 0 "" {MPLTEXT 1 0 124 "definemore(`&*`,\n&*(A::algebraic,A::algebraic,B::al gebraic)=&*(A^2,B),&*(A::algebraic,B::algebraic,B::algebraic)=&*(A,B^2 ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(C3);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,8*&^##\"\"\"\"#CF'-%#&*G6$*$)%\"AG\" \"#F'%\"BGF'F'*&^##\"\"&\"#[F'-F*6$F,F.F'F'*&^##!\"\"\"#7F'-F*6$*$)F0F /F'F.F'F'*&F9F'-F*6%F.F0F.F'F'*&^##F;\"\"'F')F0\"\"$F'F'*&^##F;F(F')F. FIF'F'*&^##F'FGF'-F*6$F?F0F'F'*&FOF'-F*6%F0F.F0F'F'*&^##F;\"#;F'-F*6$F .F,F'F'*&F9F'-F*6$F.F?F'F'*&F%F'-F*6$F0F,F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "definemore(`&*`,\n&*(A::algebraic,A::algebraic, A::algebraic)=A^3,&*(A::algebraic,(A::algebraic)^2)=A^3,&*((A::algebra ic)^2,A::algebraic)=A^3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(C3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&^##\"\"\"\"#C F'-%#&*G6$*$)%\"AG\"\"#F'%\"BGF'F'*&^##!\"\"\"#7F'-F*6$*$)F0F/F'F.F'F' *&F2F'-F*6%F.F0F.F'F'*&^##F'\"\"'F'-F*6%F0F.F0F'F'*&F2F'-F*6$F.F8F'F'* &F%F'-F*6$F0F,F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The third-o rder error term can be expressed in terms of commutators:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "CM:=(A,B)->A &* B-B &* A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CMGf*6$%\"AG%\"BG6\"6$%)operatorG%&arrowG F),&-%#&*G6$9$9%\"\"\"-F/6$F2F1!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "C3f:=I*(CM(A+2*B,CM(A,B)))/24;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$C3fG*&^##\"\"\"\"#CF(,.-%#&*G6$*$)%\"AG\"\"#F(%\"BGF (*&F1F(-F,6%F0F2F0F(!\"\"*&\"\"%F(-F,6%F2F0F2F(F(*&F1F(-F,6$*$)F2F1F(F 0F(F6*&F1F(-F,6$F0F>F(F6-F,6$F2F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(C3-C3f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Now show that this is better than the naive spl itting:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "UtA:=convert(tay lor(exp(-I*(A)*h),h=0,No),polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$UtAG,,\"\"\"F&*(^#!\"\"F&%\"AGF&%\"hGF&F&*(\"\"#F)F*F-F+F-F)*(^##F &\"\"'F&)F*\"\"$F&)F+F3F&F&*(\"#CF)F*\"\"%F+F7F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "UtB:=convert(taylor(exp(-I*(B)*h),h=0,No),pol ynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$UtBG,,\"\"\"F&*(^#!\"\"F& %\"BGF&%\"hGF&F&*(\"\"#F)F*F-F+F-F)*(^##F&\"\"'F&)F*\"\"$F&)F+F3F&F&*( \"#CF)F*\"\"%F+F7F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Uspo :=simplify(subs(h=dt,UtA &* UtB));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%%UspoG,T\"\"\"F&*(^##!\"\"\"#CF&)%#dtG\"\"&F&-%#&*G6$%\"AG*$)%\"BG\" \"%F&F&F&*(^##F&\"\"#F&)F-\"\"$F&-F06$F2*$)F5F:F&F&F&*(F8F&F;F&-F06$*$ )F2F:F&F5F&F&*(^##F*\"#7F&F,F&-F06$*$)F2F " 0 "" {MPLTEXT 1 0 35 "Udif:=simplify (subs(h=dt,Ut)-Uspo);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%UdifG,jo*& #\"\"\"\"#[F(*&)%#dtG\"\"'F(-%#&*G6$*$)%\"AG\"\"#F(*$)%\"BG\"\"%F(F(F( F(*&#F(\"#OF(*&F+F(-F/6$*$)F3\"\"$F(*$)F7FAF(F(F(F(*&#F(F-F(*&)F,F8F(- F/6$F?F7F(F(!\"\"*&#F(\"$w&F(*&)F,\"\")F(-F/6$*$)F3F8F(F5F(F(FJ*&#F(F8 F(*&FGF(-F/6$F1*$)F7F4F(F(F(FJ*&F'F(*&F+F(-F/6$FSFZF(F(F(*&#F(F-F(*&FG F(-F/6$F3FBF(F(FJ*(^##F(\"#CF()F,\"\"&F(-F/6$F3F5F(F(*(^##F(\"#7F(FcoF (-F/6$F?FZF(F(*(^##FJ\"$W\"F()F,\"\"(F(-F/6$F?F5F(F(*(F^pF(FapF(-F/6$F SFBF(F(*(FhoF(FcoF(-F/6$F1FBF(F(*(F`oF(FcoF(-F/6$FSF7F(F(*&FaoF(*&FGF( -F/6&F3F3F3F3F(F(F(*&FaoF(*&FGF(-F/6&F3F3F3F7F(F(F(*&FaoF(*&FGF(-F/6&F 3F3F7F3F(F(F(*(^##F(F-F()F,FAF(-F/6%F7F3F7F(F(*&FaoF(*&FGF(-F/6&F3F3F7 F7F(F(F(*&FaoF(*&FGF(-F/6&F3F7F3F3F(F(F(*&FaoF(*&FGF(-F/6&F3F7F3F7F(F( F(*&FaoF(*&FGF(-F/6&F3F7F7F3F(F(F(*&FaoF(*&FGF(-F/6&F3F7F7F7F(F(F(*&Fa oF(*&FGF(-F/6&F7F3F3F3F(F(F(*&FaoF(*&FGF(-F/6&F7F3F3F7F(F(F(*&FaoF(*&F GF(-F/6&F7F3F7F3F(F(F(*&FaoF(*&FGF(-F/6&F7F3F7F7F(F(F(*&FaoF(*&FGF(-F/ 6&F7F7F3F3F(F(F(*&FaoF(*&FGF(-F/6&F7F7F3F7F(F(F(*&FaoF(*&FGF(-F/6&F7F7 F7F3F(F(F(*&FaoF(*&FGF(-F/6&F7F7F7F7F(F(F(*&#F(F4F(*&)F,F4F(-F/6$F3F7F (F(F(*(^##FJFAF(F]rF(-F/6$F3FZF(F(*(F[vF(F]rF(-F/6$F1F7F(F(*(F[rF(F]rF (-F/6%F3F7F3F(F(*(F[rF(F]rF(-F/6$F7F1F(F(*(F[rF(F]rF(-F/6$FZF3F(F(*(Fb oFJF7F8F,F8FJ*(FboFJF3F8F,F8FJ*&#F(F4F(*&FguF(-F/6$F7F3F(F(FJ" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "coeff(Udif,dt,0),coeff(Udif, dt,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "C2:=coeff(Udif,dt,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2G,&*&#\"\"\"\"\"#F(-%#&*G6$%\"AG%\"BGF(F(*&#F(F)F( -F+6$F.F-F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The simple pr oduct is only second-order accurate, i.e., an " }{TEXT 19 7 "O(dt^2)" }{TEXT -1 17 " term survive if " }{TEXT 19 1 "A" }{TEXT -1 5 " and " } {TEXT 19 1 "B" }{TEXT -1 15 " don't commute." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 9 "Exercise:" }}{PARA 0 "" 0 "" {TEXT -1 105 "Look at the Campbell-Baker-Hausdorff expansion with a sm allness parameter dt to generate a Taylor series." }}{PARA 0 "" 0 "" {TEXT 19 31 "exp(A*dt) exp(B*dt) = exp(O*dt)" }{TEXT -1 8 ", where " } {TEXT 19 58 "O = A + B + 1/2 [A,B] + 1/12 ([[A,B],B] + [A,[A,B]]) + .. ." }}{PARA 0 "" 0 "" {TEXT -1 94 "Use the Taylor expansion to verify t he quoted result, and calculate the next order, i.e., the " }{TEXT 19 7 "O(dt^3)" }{TEXT -1 12 " expression." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "The next issue: can we gener ate a more complicated split-operator technique which will agree with \+ the exact answer to a higher order in dt?" }}{PARA 0 "" 0 "" {TEXT -1 87 "The answer is yes. We first define the basic split-operator techni que as a single step:" }}{PARA 0 "" 0 "" {TEXT 19 47 "U2 = exp(-I*A*dt /2) exp(-I*B*dt) exp(-I*A*dt/2)" }{TEXT -1 18 " as a function of " } {TEXT 19 8 "A, B, dt" }}{PARA 0 "" 0 "" {TEXT 19 48 "U3 = U2(A,B,g*dt) U2(A,B,(1-2*g)*dt) U2(A,B,g*t)" }{TEXT -1 38 " will work for a partic ular choice of " }{TEXT 19 1 "g" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "U2:=(A,B,h)->exp(-I*h/2*A) &* exp(-I*h*B) &* e xp(-I*h/2*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U2Gf*6%%\"AG%\"BG% \"hG6\"6$%)operatorG%&arrowGF*-%#&*G6$-F/6$-%$expG6#*(^##!\"\"\"\"#\" \"\"9&F;9$F;-F46#*(^#F9F;F " 0 " " {MPLTEXT 1 0 68 "g:='g': # a parameter to subdivide the time step dt to be optimized." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "U3:=U2 (A,B,g*h) &* U2(A,B,(1-2*g)*h) &* U2(A,B,g*h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#U3G-%#&*G6+-%$expG6#**^##!\"\"\"\"#\"\"\"%\"gGF0%\"h GF0%\"AGF0-F)6#**^#F.F0F1F0F2F0%\"BGF0F(-F)6#**F,F0,&F0F0*&F/F0F1F0F.F 0F2F0F3F0-F)6#**F7F0F " 0 "" {MPLTEXT 1 0 35 "convert(taylor(U3,h=0,No),polynom);" }}{PARA 8 "" 1 " " {TEXT -1 76 "Error, (in D/procedure) index out of range: function ta kes only 0 arguments\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "The di rect approach won't work. We need to do what was done before, namely g enerate the Taylor expansions for the individual pieces, and put them \+ together." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "UtA:=convert(t aylor(exp(-I*(A)*h/2),h=0,No),polynom);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "UtB:=convert(taylor(exp(-I*(B)*h),h=0,No),polynom);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$UtAG,,\"\"\"F&*(^##!\"\" \"\"#F&%\"AGF&%\"hGF&F&*(\"\")F*F,F+F-F+F**(^##F&\"#[F&)F,\"\"$F&)F-F5 F&F&*(\"$%QF*F,\"\"%F-F9F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$UtBG, ,\"\"\"F&*(^#!\"\"F&%\"BGF&%\"hGF&F&*(\"\"#F)F*F-F+F-F)*(^##F&\"\"'F&) F*\"\"$F&)F+F3F&F&*(\"#CF)F*\"\"%F+F7F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "U2:=unapply(UtA &* UtB &* UtA, A,B,h):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The naive implementation is given below. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# U3:=U2(A,B,g*h) &* U2 (A,B,(1-2*g)*h) &* U2(A,B,g*h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "constants:=constants,g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%* constantsG6+%&falseG%&gammaG%)infinityG%%trueG%(CatalanG%%FAILG%#PiG%# dtG%\"gG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "#U3:=simplify( \+ subs(h=dt,U2(A,B,g*h) &* U2(A,B,(1-2*g)*h) &* U2(A,B,g*h)) ):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "The above line would at least make use of the fact that " }{TEXT 19 2 "dt" }{TEXT -1 242 " is a constant , and simplify operator products. It is, however, too cumbersome for M aple to evaluate within a reasonable amount of memory and time. The tr ick is to build up the expression gradually, and to keep only the lowe st four orders in " }{TEXT 19 2 "dt" }{TEXT -1 86 " for each factor as one builds the expression. Note that back-and-forth substitutions " } {TEXT 19 2 "dt" }{TEXT -1 4 " vs " }{TEXT 19 1 "h" }{TEXT -1 45 " are \+ needed. Taylor expansion will work with " }{TEXT 19 1 "h" }{TEXT -1 15 ", and not with " }{TEXT 19 2 "dt" }{TEXT -1 50 ". Non-commutative \+ product simplification requires " }{TEXT 19 2 "dt" }{TEXT -1 41 ", so \+ that it is recognized as a constant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "U3a:=convert(taylor(subs(dt=h,simplify(subs(h=dt,U2(A ,B,g*h)))),h=0,No),polynom);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$U3a G,,\"\"\"F&*&,&*(^#!\"\"F&%\"AGF&%\"gGF&F&*(F*F&F-F&%\"BGF&F&F&%\"hGF& F&*&,**&#F&\"\"#F&*&)F-F5F&-%#&*G6$F,F/F&F&F+*(F5F+F/F5F-F5F+*(F5F+F,F 5F-F5F+*&#F&F5F&*&F7F&-F96$F/F,F&F&F+F&)F0F5F&F&*&,0*(^##F&\"\"%F&)F- \"\"$F&-F96%F,F/F,F&F&*(^##F&\"\"'F&)F/FJF&FIF&F&*(^##F&\"\")F&FIF&-F9 6$F/*$)F,F5F&F&F&*(FFF&FIF&-F96$F,*$)F/F5F&F&F&*(FFF&FIF&-F96$FgnF,F&F &*(FSF&FIF&-F96$FXF/F&F&*(FNF&)F,FJF&FIF&F&F&)F0FJF&F&*&,>*&#F&\"#;F&* &)F-FHF&-F96%FXF/F,F&F&F&*&#F&\"#kF&*&FhoF&-F96$FXFXF&F&F&*&FeoF&*&Fho F&-F96$FXFgnF&F&F&*&#F&\"#7F&*&FhoF&-F96$F,*$FQF&F&F&F&*&#F&\"#[F&*&Fh oF&-F96$*$F`oF&F/F&F&F&*&FeoF&*&FhoF&-F96$FgnFXF&F&F&*&FfpF&*&FhoF&-F9 6$F[qF,F&F&F&*&#F&\"#'*F&*&FhoF&-F96$FbqF,F&F&F&*&FTF&*&FhoF&-F96%F,Fg nF,F&F&F&*&FeoF&*&FhoF&-F96%F,F/FXF&F&F&*(\"#CF+F/FHF-FHF&*(\"$#>F+F,F HF-FHF&*&F]qF&*&FhoF&-F96$F/FbqF&F&F&*&F\\rF&*&FhoF&-F96$F,FbqF&F&F&F& )F0FHF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "U3b:=convert(t aylor(subs(dt=h,simplify(subs(h=dt,U2(A,B,(1-2*g)*h)))),h=0,No),polyno m);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$U3bG,,\"\"\"F&*&,**&^#!\"\"F &%\"BGF&F&*(^#\"\"#F&F,F&%\"gGF&F&*(F.F&%\"AGF&F0F&F&*&F*F&F2F&F&F&%\" hGF&F&*&,:*(F/F&)F0F/F&-%#&*G6$F,F2F&F+*&#F&F/F&-F:6$F2F,F&F+*(F/F&)F, F/F&F8F&F+*(F/F&)F2F/F&F0F&F&*&F/F+F2F/F+*(F/F&F9F&F0F&F&*&F/F+F,F/F+* &#F&F/F&F9F&F+*(F/F&F>F&F0F&F&*(F/F&FCF&F8F&F+*(F/F&FAF&F0F&F&*(F/F&F8 F&F>F&F+F&)F4F/F&F&*&,Z*(F*F&)F,\"\"$F&F0F&F&*(^#FRF&-F:6$*$FAF&F2F&F8 F&F&*(^##!\"$F/F&FUF&F0F&F&*(^##Fen\"\"%F&-F:6$F,*$FCF&F&F0F&F&*(F*F&F jnF&)F0FRF&F&*(FYF&-F:6$F2FWF&F0F&F&*(^##!\"%FRF&FQF&F^oF&F&*(F.F&FQF& F8F&F&*(FTF&F`oF&F8F&F&*(F*F&-F:6$F\\oF,F&F^oF&F&*(^#!\"#F&-F:6%F2F,F2 F&F^oF&F&*(FTF&F^pF&F8F&F&*(FgnF&FioF&F0F&F&*(^##FRF/F&FjnF&F8F&F&*(FY F&F^pF&F0F&F&*(F\\pF&F`oF&F^oF&F&*(FcpF&FioF&F8F&F&*(F*F&)F2FRF&F0F&F& *(F.F&FipF&F8F&F&*(FcoF&FipF&F^oF&F&*&^##F&\"\")F&FioF&F&*&^##F&FinF&F UF&F&*&FaqF&F`oF&F&*&FaqF&F^pF&F&*&^##F&\"\"'F&FQF&F&*&F]qF&FjnF&F&*&F fqF&FipF&F&*(F\\pF&FUF&F^oF&F&F&)F4FRF&F&*&,hs*&#F&\"#'*F&-F:6$*$FipF& F2F&F&*&#F&\"#[F&-F:6$F,FdrF&F&*&FbqF&*&FbrF&F8F&F&F&*&#F/FRF&*&FhrF&F ^oF&F&F+*&#F&\"#7F&-F:6$*$FQF&F2F&F&*&#F&\"#;F&-F:6%F\\oF,F2F&F&*&FfsF &-F:6%F2F,F\\oF&F&*&F^qF&-F:6%F2FWF2F&F&*&FfrF&-F:6$FdrF,F&F&*&FfsF&-F :6$F\\oFWF&F&*&FbqF&*&)F0FinF&-F:6$F\\oF\\oF&F&F&*&F`sF&-F:6$F2FdsF&F& *&#FinFRF&*&FhtF&FbsF&F&F&*&FhtF&FdtF&F&*&FgqF&*&FhtF&-F:6$F2FdrF&F&F& *&#F/FRF&*&F\\uF&F0F&F&F+*&#F_qFRF&*&F\\uF&F^oF&F&F+*(F/F&F\\uF&F8F&F& *&F_uF&*&FhtF&F\\uF&F&F&*&#F&F_qF&*&FitF&F0F&F&F+*&FdpF&*&F[tF&F8F&F&F &*(FRF+F,FinF0F&F+*(FhqF+F2FinF0FRF+*(\"#CF+F2FinF0F&F+*&)F,FinF&F8F&F &*(F_qF+F2FinF0F/F&*&#FRF_qF&*&FitF&F8F&F&F&**FinF&FRF+F,FinF0FRF+*&#F &FRF&*&FduF&F^oF&F&F+*&#F&F/F&*&FdtF&F0F&F&F+*(F/F&FdtF&F^oF&F+*&FhtF& FhsF&F&*&#F&FhqF&*&FatF&F0F&F&F+*&#F/FRF&*&FatF&F^oF&F&F+*&FdpF&*&FhsF &F8F&F&F&*(F/F&FhsF&F^oF&F+*&#F&FRF&*&FbrF&F^oF&F&F+*&#F&F/F&*&FhsF&F0 F&F&F+*&#F&F/F&*&FatF&F8F&F&F&*&FdpF&*&FdtF&F8F&F&F&*&FbqF&*&FduF&F8F& F&F&*(FRF&F^tF&F8F&F&*&#F&F/F&*&F[tF&F0F&F&F+*(F/F&F[tF&F^oF&F+*(F/F&F bsF&F8F&F&*&#F_qFRF&*&FbsF&F^oF&F&F+*&#F/FRF&*&FbsF&F0F&F&F+*&#F&F/F&* &FitF&F^oF&F&F+*(FinF&F^tF&F^oF&F+*&#F&FasF&*&FduF&F0F&F&F+*&#F&FasF&* &FbrF&F0F&F&F+*&F^tF&F0F&F+*&FdpF&*&-F:6$FWF\\oF&F8F&F&F&*(F/F&FfzF&F^ oF&F+*&#F&F/F&*&FfzF&F0F&F&F+*&#F&FhqF&*&FhrF&F0F&F&F+*&FgxF&*&FhrF&F8 F&F&F&*&\"$#>F+F2FinF&*&FgvF+F,FinF&*&FgqF&*&FhtF&FbrF&F&F&*&#F&\"#kF& FitF&F&*&#F&FRF&*&FhtF&FatF&F&F&*&Fj[lF&*&FhtF&FhrF&F&F&*(F/F&FhtF&F^t F&F&*&FhtF&F[tF&F&*&FhtF&FfzF&F&*&FfsF&FfzF&F&*(FasF+F2FinF0FinF&*&F`r F&FduF&F&**F/F&FRF+F,FinF0FinF&F&)F4FinF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "U3p:= expand(subs(h=dt,U3a) &* subs(h=dt,U3b) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "U3p:=convert(taylor(subs( dt=h,U3p),h=0,No),polynom):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "U3p:= expand(subs(h=dt,U3p) &* subs(h=dt, U3a)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "U3:=convert(taylor(subs(dt=h,U3p),h=0,No) ,polynom):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "Note that this rel atively fast computation relies on the Taylor truncations. Try it with out them and see memory and time gobbled up." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "coeff(U3,h,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*&^#!\"\"\"\"\"%\"AGF'F'*&F%F'%\"BGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Ut;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,\"\"\"F$*( ^#!\"\"F$,&%\"AGF$%\"BGF$F$%\"hGF$F$*&#F$\"\"#F$*&,**$)F)F.F$F$-%#&*G6 $F)F*F$-F46$F*F)F$*$)F*F.F$F$F$)F+F.F$F$F'*(^##F$\"\"'F$,2*$)F)\"\"$F$ F$-F46$F1F*F$-F46%F)F*F)F$-F46$F)F8F$-F46$F*F1F$-F46%F*F)F*F$-F46$F8F) F$*$)F*FBF$F$F$)F+FBF$F$*&#F$\"#CF$*&,B-F46&F)F)F)F)F$-F46&F)F)F)F*F$- F46&F)F)F*F)F$-F46&F)F)F*F*F$-F46&F)F*F)F)F$-F46&F)F*F)F*F$-F46&F)F*F* F)F$-F46&F)F*F*F*F$-F46&F*F)F)F)F$-F46&F*F)F)F*F$-F46&F*F)F*F)F$-F46&F *F)F*F*F$-F46&F*F*F)F)F$-F46&F*F*F)F*F$-F46&F*F*F*F)F$-F46&F*F*F*F*F$F $)F+\"\"%F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Now we verify \+ that the time evolution operator expressions agree in their orders 0 t o 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "DIF:=simplify(Ut-U3 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "coeff(DIF,h,0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "coeff(DIF,h,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "coeff(DIF,h,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "DIF3:=coeff(DIF,h,3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%DIF3G,R*&^##\"\"\"\"#CF)-%#&*G6$*$)%\"AG\"\"#F)%\"BGF)F)*(^#F 1F))%\"gGF1F)-F,6%F2F0F2F)F)*(^##F)F1F))F6\"\"$F)-F,6%F0F2F0F)F)*(^##! \"\"\"\"%F)F6F)-F,6$F2F.F)F)*(^#FCF)F6F)F7F)F)*(FHF)F5F)F>F)F)*(FAF)F6 F)F+F)F)*(F:F)F6F)-F,6$*$)F2F1F)F0F)F)*&^##FC\"#7F)-F,6$F0FNF)F)*&F'F) FEF)F)*(FHF)F5F)FTF)F)*(FHF)FF)F)*&FQF)F>F)F)*(F:F)F6F)FTF) F)*(FAF)F " 0 "" {MPLTEXT 1 0 37 "simplify(subs(g =1/(2-2^(1/3)),DIF3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "This completes the proof that a fo urth-order accurate time evolution operator can be found." }}{PARA 0 " " 0 "" {TEXT -1 192 "In the numerical implementation one needs to ask \+ the question whether this algorithm is economical. The improved accura cy comes at the price of a substantial number of additional calculatio ns." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "70 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }