{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 "" -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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 19 "Wigner distribution" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 184 "The Wign er function is the basic quantity to describe ensembles in a phase-spa ce formulation of quantum mechanics. It is defined as a paricular Four ier transform of the density matrix." }}{PARA 0 "" 0 "" {TEXT 19 71 "f (q,p) = int(exp(-I*p*s)*psi(q+s/2)*psi^*(q-s/2),s=-infinity..infinity) " }}{PARA 0 "" 0 "" {TEXT -1 58 "(the integral corresponds to an inver se Fourier transform)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Let us start with the simplest examples, such as harmo nic oscillator eigenstates:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart; with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "fx:=x*exp(-x^2/2); #chose the unnormalized state here" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "No:=1/sqrt(int((fx)^2,x=-inf inity..infinity));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:= unapply(No*fx,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(ps i(x)^2,x=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "rho:=psi(x+s/2)*psi(x-s/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fW:=invfourier(rho,s,p);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "Rp:=int(fW,x=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Rx:=int(fW,p=-infinity..infinity); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(Rx,x=-3..3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int(Rp,p=-infinity..infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int(Rx,x=-infinity..i nfinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot3d(fW,x=-3 ..3,p=-3..3,axes=boxed,shading=zhue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "Nothing particularly exciting happens here for the groun d state. For the first excited state we have something non-classical: \+ the Wigner function is not positive definite!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Exercise 1:" }}{PARA 0 " " 0 "" {TEXT -1 181 "Choose different excited states of the harmonic o scillator and calculate the Wigner function. How is the number of node s in the wavefunction related to the structure in phase space?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 35 " What do we obtain for a wavepacket?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 83 "We choose a Gaussian wavepacket centered at the origin with an average momentum of " }{TEXT 19 2 "p0" }{TEXT -1 131 ". We pick a simple width (which controls the width of the mome ntum distribution for all times). We compute the answer at time zero. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "p0:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "fx:=exp(-(x)^2/2)*exp(I*p0*x); #chose the unnormalized state here" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "#fx:=1/(1+x^2)*exp(I*p0*x); # the Lorentzian wavepacket doesn't work. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "No:=1/sqrt(int(abs(fx)^ 2,x=-infinity..infinity));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi:=unapply(No*fx,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(abs(psi(x))^2,x=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "assume(x,real,s,real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "rho:=simplify(expand(psi(x+s/2)*conjugate(psi(x- s/2))));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fW:=invfourier( rho,s,p);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Rp:=int(fW,x=- infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Rx: =int(fW,p=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot([Re(Rp),Im(Rp)],p=-3..3,color=[red,blue]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Something went wrong. The Wigner function should be real-valued." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " int(Rx,x=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "fW:=simplify(expand(int(exp(-I*p*s)*rho,s=-infinity..infinity) ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(p=1,x=1,%);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot3d(fW,x=-3..3,p=-3..3,a xes=boxed,shading=zhue);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "The \+ Wigner function for a Gaussian wavepacket state is quite straightforwa rd, and is easily interpreted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 257 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 68 "Displace the Gaussian wavepacket and re-compute the Wigner functio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 11 "Ex ercise 3:" }}{PARA 0 "" 0 "" {TEXT -1 142 "Broaden the Gaussian wavepa cket in coordinate space and re-compute the Wigner function. Make your observation about the momentum distribution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "We explore now th e time-dependent Gaussian wavepacket. The width parameter " }{TEXT 19 1 "a" }{TEXT -1 69 " determines the spatial extent and momentum spread about the average." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a:=1 ; p0:='p0';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "psi := unap ply(sqrt(a)*1/Pi^(1/4)*exp(-1/2*(x-I*a^2*p0)^2/(a^2+t*I)-1/2*a^2*p0^2) /(sqrt(-a^2-I*t)),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "as sume(t>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(abs(psi(x ))^2,x=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "int(evalc(simplify(abs(psi(x))^2)),x=-infinity..infinity);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "rho:=simplify(expand(psi(x+s /2)*conjugate(psi(x-s/2))));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "fW:=simplify(expand(int(exp(-I*p*s)*rho,s=-infinity..infinity))) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fW:=evalc(fW);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "p0:=1;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "fW1:=subs(t=1,fW);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "fW2:=subs(t=2,fW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fW3:=subs(t=3,fW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Rx1:=int(fW1,p=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Rx2:=int(fW2,p=-infinity..infinity) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Rx3:=int(fW3,p=-infini ty..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot([Rx1 ,Rx2,Rx3],x=-5..10,color=[red,blue,green]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Rp1:=int(fW1,x=-infinity..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Rp2:=int(fW2,x=-infinity..infinity) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Rp3:=int(fW3,x=-infini ty..infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "The momentum d istribution remains constant! This is required, since the wavepacket u ndergoes free dispersion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "plots[animate3d](fW,x=-5..15,p=-4..4,t=0..4,axes=boxed,grid=[50,5 0],shading=zhue,style=patchcontour,frames=20);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "The apparent ripples at later times are an artefact \+ of the graphing algorithm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 11 "Exercise 4:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Observe what happens when the spatial width parameter " } {TEXT 19 1 "a" }{TEXT -1 27 " is increased or decreased." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "34 0 2" 69 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }