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 coherent state).  Also given by the following animated GIF.



2)  AVI file of WF for a linear combination of ground and first excited SHO states:  ( |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], 1/(2^1/3πB) AiryAi[2^2/3(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 perspective.)

    ^[1] M V Berry and N L Balazs, Am J Phys 47 (1979) 264-267.