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{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 }