Time Dependent Wigner Functions
This page (© Thomas
Curtright 2011 & 2012) is based on concepts developed in Cosmas K Zachos, David B
Fairlie, and Thomas L Curtright, Quantum
Mechanics in Phase Space: An Overview with Selected Papers, World Scientific, 2005, and
references cited therein. For the early history of formulating quantum
mechanics in phase space, see http://arxiv.org/abs/1104.5269.
1) AVI file of WF for the SHO
ground state displaced from the origin of phase space (i.e. a
Also given by the following animated GIF.
2) AVI file of WF for a linear combination of ground and first excited SHO
|0> + |1> ) / √2. Also given by the following animated GIF, as well
as one for the same linear combination after being displaced from
the origin of phase space.
3) Truncated view of time
development for the "Airy front", 1/(21/3πB) AiryAi[22/3(Bx+p²/B²-2Bpt)]
for constant B, a pure state Wigner distribution whose shape in x
and p is invariant as time progresses, but whose features
accelerate to the right even while propagating freely!
Nevertheless, Ehrenfest can rest in peace in this force-free
situation because the state is both non-normalizable and has an
undefined (infinite) <x> for all t. Moreover, the
momentum distribution obtained by integrating over all x is constant. So, to the
extent that it can be defined, <p>=0 for all t despite the
apparent acceleration of the front. (I thank
M Vanhuss for early emphasis on the advantages of an overhead
M V Berry and N L Balazs, Am J Phys 47 (1979) 264-267.