{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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 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 24 "Phase Shift Calculations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "We te st a few methods for calculating higher-L phase shifts by quantum, and by WKB methods." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "We use a potential for which some reference results are g iven in Joachain (page 218, Table 9.7)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart; Digits:=14: with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "We start by defining asymptotic solutions and the ir derivatives." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pi:=eval f(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#piG$\"/)*e`EfTJ!#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "F:=(k,r,L)->sqrt(0.5*pi*k*r) *BesselJ(L+1/2,k*r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6%%\"k G%\"rG%\"LG6\"6$%)operatorG%&arrowGF**&-%%sqrtG6#**$\"\"&!\"\"\"\"\"%# piGF69$F69%F6F6-%(BesselJG6$,&9&F6#F6\"\"#F6*&F8F6F9F6F6F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "G:=(k,r,L)->sqrt(0.5*pi*k*r) *(-1)^L*BesselJ(-L-1/2,k*r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG f*6%%\"kG%\"rG%\"LG6\"6$%)operatorG%&arrowGF**(-%%sqrtG6#**$\"\"&!\"\" \"\"\"%#piGF69$F69%F6F6)F59&F6-%(BesselJG6$,&F;F5#F6\"\"#F5*&F8F6F9F6F 6F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Fp:=(K,r,L)->eva lf(eval(diff(F(K,s,L),s),s=r));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# FpGf*6%%\"KG%\"rG%\"LG6\"6$%)operatorG%&arrowGF*-%&evalfG6#-%%evalG6$- %%diffG6$-%\"FG6%9$%\"sG9&F;/F;9%F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Gp:=(K,r,L)->evalf(eval(diff(G(K,s,L),s),s=r));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GpGf*6%%\"KG%\"rG%\"LG6\"6$%)operat orG%&arrowGF*-%&evalfG6#-%%evalG6$-%%diffG6$-%\"GG6%9$%\"sG9&F;/F;9%F* F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "PhSh:=proc() local \+ k,SE,sol,uS,u1S,Rcut,uR,u1R,N,IC,eta,MatchEq,BoA; global V,L,pi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "if type(args[1],numeric) then k:=ar gs[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SE:=diff(u(r),r$2)+(k^2-2 *V-L*(L+1)/r^2)*u(r)=0; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "eta:=(1 E-3)*1.01^L; IC:=u(eta)=eta^(L+1),D(u)(eta)=(L+1)*eta^L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "sol:=dsolve(\{SE,IC\},numeric,method=dverk78, output=listprocedure,maxfun=0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " uS:=eval(u(r),sol):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Rcut:=5E1;# \+ this is somewhat arbitrary!!!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "u1 S:=eval(diff(u(r),r),sol):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "uR:=u S(Rcut);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "u1R:=u1S(Rcut);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "MatchEq:=u1R/uR=(Fp(k,Rcut,L)+BoA*G p(k,Rcut,L))/(F(k,Rcut,L)+BoA*G(k,Rcut,L));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "arctan(solve(MatchEq,BoA));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "elif type(args [1],list) then N:=nops(args[1]); [seq(PhSh(args[1][j]),j=1..N)];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 222 "Note that we use \+ a high-order method to integrate the solution out! We should be testin g for a given potential which value of Rcut can be used! The potential is passed as a global variable, as is the partial-wave number L." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "V:=-125/(r^2+1)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$*&\"$D\"\"\"\",&*$)%\"rG\"\"#F(F(F(F (!\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "K:=5;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"\" &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "PhSh(K);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!/[]GG')[Y!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Joachain list the value as 8.96" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "We get only answers within (-pi,pi)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "3*pi+%;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"/*=zn\"*)f*)!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "We will try to understand why the actual phase shift is so lar ge by comparing the solution against the free solution. 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potential is dominated by the centrifugal potential for distances beyo nd r=5. Yet, an accurate determination of the phaseshift requires inte gration of the Schroedinger equation to larger distances (50 units in \+ our code). Change Rcut in the procedure PhSh to values closer to Rcut= 5, and observe what happens to the phaseshift.Then change to values Rc ut<5 and make your observation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 266 "To under stand the full value of the phaseshift it is worth comparing the solut ion for the free Schroedinger equation to the full solution, both star ted with the same initial condition. This will also illustrate the rol e of the normalization in the asymptotic regime." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We define integrators for the free problem SE0 and for the full problem SE." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "SE0:=proc(Rout) local k,SE,sol,uS,u1S,uR,u1 R,N,IC,eta; global V,L,pi,K;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=K ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "SE:=diff(u(r),r$2)+(k^2-L*(L+1 )/r^2)*u(r)=0; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "eta:=(1E-3)*1.01 ^L; IC:=u(eta)=eta^(L+1),D(u)(eta)=(L+1)*eta^L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "sol:=dsolve(\{SE,IC\},numeric,method=dverk78,output=l istprocedure,maxfun=0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "uS:=eval (u(r),sol):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "uS(Rout);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "SE1:=proc(Rout) local k,SE,sol,uS,u1S,uR,u1R,N,IC,eta; global \+ V,L,pi,K;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=K;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "SE:=diff(u(r),r$2)+(k^2-2*V-L*(L+1)/r^2)*u(r)=0; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "eta:=(1E-3)*1.01^L; IC:=u(eta) =eta^(L+1),D(u)(eta)=(L+1)*eta^L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "sol:=dsolve(\{SE,IC\},numeric,method=dverk78,output=listprocedure, maxfun=0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "uS:=eval(u(r),sol):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "uS(Rout);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Note that the only difference is in the definition of the Schroedinger equation ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "PL1:=plot(SE1(R),R=0..8,color=red,numpoints=10 00): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "PL2:=plot(0.003*SE 0(R),R=0..8,color=blue,numpoints=1000):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(PL1,PL2);" }}{PARA 13 "" 1 "" {GLPLOT2D 682 322 322 {PLOTDATA 2 "6&-%'CURVESG6$7b[o7$$\"/oxwnF=E!#<$!/Xvr3d1&)!#J7 $$\"/Nb`NbO_F*$\"/\"*Gg\\DcX!#L7$$\"/+LI.$[&yF*$\"/_VY_p[B!#K7$$\"/2r5 2JZ5!#;$\"/5\")4Lj>8F-7$$\"/g1mg'4d\"F=$\"/IHjw1.:!#I7$$\"/9U@9i%4#F=$ \"/WEaOWG%)FE7$$\"/@8K@$>9$F=$\"/j(Hp%\\Z#*!#H7$$\"/G%G%GC*=%F=$\"/S&e )z$Hq%!#G7$$\"/SEkU'QG'F=$\"/?Q*GvVm&!#F7$$\"/do&o&[y$)F=$\"/#Hk4Xx!G! 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The very different effective p otential at small r is responsible for these different amplitudes." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "The matc hing algorithm that extracted the phase shift from the solution SE1 (t hat is what happened effectively in PhSh) divided out the normalizatio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 256 "By inspecting the graph one observes how the blue curve (free solution) \+ misses 1.5 oscillations (the red curve is calculated including an attr active potential!). These 1.5 oscillations correspond to a phase shift of about 3*pi (the accurate value is 8.97)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 202 "Note that there is a glitch in the solution. For this reason professionals use special-purpose algor ithms, such as a method developed by De Vogelaere, and generalized in \+ J. Computational Physics 3, 322." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "An aside:" }}{PARA 0 "" 0 "" {TEXT -1 138 "Can we measure the phase shift directly? We can determine the zero cr ossings using a bisection algorithm. It is made up of two procedures: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "bis:=proc(f,a,b) local \+ fa,fb,xh,fh; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "fa:=f(a): fb:=f(b ): if f(a)*f(b)<0 then xh:=0.5*(a+b): fh:=f(xh): if fh*fa<0 then [a,xh ]; else [xh,b]; fi: else RETURN(\"No simple root\") fi: end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bis(SE1,4.,4.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"$0%!\"#$\"#T!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "BS:=proc(f,a,b) local xr,i,xnew;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "xr:=bis(f,a,b);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for i from 1 to 100 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "xnew:=bis(f,xr[1],xr[2]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "if abs(xnew[1]-xnew[2])<1E-6 then RETURN(0.5*(xnew[1] +xnew[2])); fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "xr:=xnew; od:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "RETURN(\"ran out of 100 steps in B iSect\"); end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Now go for a ma tching pair of zero crossings:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xR:=BS(SE1,7.2,7.3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xRG $\"/I35s$*Rs!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xB:=BS(S E0,7.7,7.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xBG$\"/I^]hswx!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "-K*(xR-xB)+2*pi;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/'H$4+8n*)!#8" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 347 "We have an accurate answer by extracting at this \+ early distance! We added 2*pi to account for the missed oscillation, a nd it it K*(Dr) which corresponds to the phase shift. One has to be ca reful about the sign (or order of subtraction). Find the distance at w hich the phase shift will agree to 3 digits with Joachain's result (et a_5=8.96 radians)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can also explore some other, simpler expressions for \+ the phaseshift. We start with the WKB result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "etaSC:=proc(L) local aL,val,kL; global K,pi; kL :=R->evalf(eval(sqrt(K^2-2*V-((L+1/2)/r)^2),r=R)); aL:=fsolve(kL(R),R= 1E-14..10)+1E-14; val:=evalf(Int(kL(r)-K,r=aL..infinity,epsilon=1E-5)) -K*aL+(L+1/2)*pi/2; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " etaSC(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/4z()Rw`*)!#8" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "This is remarkably close to the f ull qm answer. It is also remarkable that the WKB expression which is \+ based upon the area under a curve gives an absolute estimate for the p hase shift." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "The high-energy approximation is probably not justified at the energy chosen, but let us calculate it anyway:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "HEM:=proc(Lc) local r0; global K,V; r0:=(Lc+0. 5)/K;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "-2/K*evalf(Int(eval(V*r,r= r0+s^2)/sqrt(s^2+2*r0),s=0..infinity,epsilon=1E-12));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "HEM(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"//()y]Uwf!#8" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 262 "The area under the curve in princ iple is capable of giving us the correct absolute value of the phase s hift. The bad news is that it doesn't come out very well. This is not \+ surprising, since the wavenumber chosen is not that high compared to t he potential depth." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 93 "We can check this out by increasing K, and keeping our \+ fingers crossed that all methods work:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "K:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "HEM(L);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"/)G.!*)R/m!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "etaSC(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/T:_8, =m!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "It looks like the high- energy limit approximation to the WKB result is accurate in this case. Let us verify the answer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "PhSh(K)+2*pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/N " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 16 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }