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# Dynamics

Since , for each , contains exactly the same information as the HILBERT space vector it is possible to give a complete formulation of quantum mechanics by means of phase space distribution functions only, that is without referring to quantum state vectors or wave functions. In such a framework describes the state of the system at time , whereas the time evolution of is determined by the equation of motion

 (A.50)

which takes the role normally occupied by the SCHRÖDINGER equation in the conventional formulation of quantum mechanics. Here, is the transform of the Hamiltonian; it is obtained by first ordering the Hamiltonian according to the kernel function , as described in section A.1, and then replacing the operators , with the scalars , . COHEN was the first to formulate the equation of motion (A.80) [Coh66]; a derivation starting from the VON NEUMANN equation can be found in [Lee95].

As an example, for the special case of the WIGNER distribution function equation (A.80) becomes

 (A.51)

Obviously, the equations of motion (A.80) and (A.81) are quite awkward to work with. This is the most important reason why in practical applications quantum mechanics hardly ever is discussed with distribution functions completely replacing quantum states and wave functions. Rather, even when the final task is to obtain distribution functions, typically the well-established version of quantum mechanics is used, i.e. one starts by solving -- numerically, if necessary -- the SCHRÖDINGER equation and then inserts the solution into equation (A.13) or (A.74), for instance, in order to compute the desired .

But in addition to the computation of distribution functions, the equation of motion for WIGNER's function is of importance for the investigation of the semiclassical approximation with . For Hamiltonians of the type equation (A.81) can be ordered with respect to powers of :

 (A.52)

This equation was first derived by WIGNER in 1932 [Wig32]. Using the POISSON bracket , it can be rewritten in the form
 (A.53)

with as usual denoting terms that are at least quadratic in and that give rise to the quantum mechanical corrections to the classical phase space distribution function as obtained from the classical LIOUVILLE equation
 (A.54)

Therefore in the classical limit the equation of motion (A.83) for formally becomes the LIOUVILLE equation (A.84); this makes WIGNER's function a useful tool for the investigation of the correspondence issue. Furthermore, for potentials that do not contain powers of exceeding 2, the dynamics of is completely classical, since in this case the equations (A.83) and (A.84) coincide. But note that itself depends on , such that the similarity of these quantum mechanical and classical equations of motion is formal in the first place, and the details of obtaining equation (A.84) from (A.83) in the classical limit are nontrivial in general.

I return to equations (A.82) and (A.83) in section A.5 when I discuss the interpretation of as a (quasi-) probability distribution function in quantum phase space.

Next: On the Interpretation as Up: Quantum Phase Space Distribution Previous: The HUSIMI Distribution Function   Contents
Martin Engel 2004-01-01