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16 Hermite Polynomials
cohsx = cohs * obj.wave;
% coh. s. coordinate repres.
y = abs(cohsx);
% absolute value for visualisation
c = angle(cohsx); % value for color coding the phase
surface([x;x],[y;y],[z;z],[c;c],’facecolor’,’none’,...
’edgecolor’,’flat’,’linewidth’,5), colorbar
caxis([-pi,pi]);%surface object - color range for phase
xlim([x(1),x(end)])
colormap jet
gsurf=get(gca,’Children’);
% handle surface object
k = k+1;
F(k) = getframe;
% 1st frame for animation
for al = at(2:end)
% frames for movie
alp = al * ones(size(nf));
alp = [1,alp];
cohs = alp./sqrt(ng);
cohs = cumprod(cohs);
cohs = exp(-1/2 * abs(alpha).^2). * cohs;
cohsx = cohs * obj.wave;
y = abs(cohsx);
% update absolute value and
c = angle(cohsx);
% color for phase
gsurf.YData = [y;y]; % update of y-value
gsurf.CData = [c;c]; % update of color coding
k = k+1;
F(k) = getframe;
end
movie(F,3) % see MATLAB documentation for details
The visualization is based on the MATLAB function surface([x;x],[y;y],
[z;z],[c;c],...). Because we have a line object we create matrices
by [x;x],... necessary for surface objects. The color coded phase is
created via [c;c] and the update in each time step via gsurf.YData =
[y;y]; gsurf.CData = [c;c]; with gsurf the corresponding object
handle.
The Wigner Function in Phase Space
In classical dynamics there is no uncertainty principle and thus it is possible to know
the position and the momentum of a particle at the same time to an (theoretically)
arbitrary precision. This would not be possible in quantum dynamics. Therefore,
there exists no phase space representation in quantum dynamics of canonically
conjugate variables. Nevertheless, the Wigner function is constructed to come close
to this picture.
The density operator of a pure state is ˆ
ρ = |ψ and in coordinate representation
ˆ
ρ| ˜
x = ψ(x)
†
· ψ( ˜
x,
(16.21)