Abstract It is demonstrated with a particle in the one-dimensional configuration space (OCS) that knowledge of the wave function supposes more than statistical restrictions on the results of measurements. In particular, apart from the probability (density) field in the OCS, the wave function also suggests the existence of two fields that predict two (equiprobable) particle momentum values for each OCS point, and the average of these two momenta at each point is related only to the wave function phase, while their difference (coinciding with the Bohm quantum-mechanical potential) is related only to the wave function amplitude. An analogue of the Heisenberg uncertainty relation is obtained for both fields.
The phase-space formulation of quantum scattering is discussed and the asymptotic quantities describing potential scattering in one and three dimensions (transmittances, cross sections) are analyzed. It is found that under certain conditions the transmittance in time-dependent scattering only depends on the momentum distribution of the incident packet, being independent of its particular shape in coordinate space. For three-dimensional scattering, generalized cross sections are defined for momentum coherences and a physical interpretation of the generalized optical theorem is proposed.
The classical limit of two quantum mechanical oscillators on a non-commutative configuration space has been recently studied based on the anti-Wick quantization scheme which makes use of completely positive maps constructed by means of coherent states. The limits corresponding to the removal of configuration-space non-commutativity and position-momentum non-commutativity do not commute. While reducing to a commutative configuration-space and then to classical position and momentum yields a classical theory on a
4-dimensional phase-space, the inverted sequence of limits does instead provide a classical theory on a
2-dimensional phase-space. Moreover, if the free dynamics of the two non-commutative quantum oscillators is also taken into account, in the first case one again obtains the dynamics of two classical free oscillators dynamics; instead, in the second case, one loses completely track of the dynamics and reduces to a purely kinematical
1-dimensional description. We herewith explain these results by examining the fate of the phase-space localisation properties of the Wigner functions of coherent states under the various limits.