Current and quasi-probability phase-space distributions

Quantum mechanical current has many features in common with classical current, but sometimes its behaviour is unusual and leads to many interesting peculiarities as has been noted by many authors. We show that many peculiarities often occur for wave functions that exhibit multiple trajectories or regions in phase space. Such wave functions have been called 'multi-part' wave functions and the trajectories are the quantum currents of each part. We contrast the behaviour of quantum current with classic current, and we pinpoint some of the curiosities. We argue and show that, in general, multi-part wave functions produce highly oscillating currents, which in itself is not peculiar, but what is peculiar is that the current can range outside the range of the momentum wave function in a variety of different situations. However, we also show that for some multi-part wave functions, the quantum mechanical current is not peculiar and is totally consistent with what one would expect for classical currents. We give the conditions for which this occurs, which are very restrictive. We also explore the interpretation of quantum mechanical current as a first conditional moment of a quantum quasi distribution, and contrast it with the conditional moment that is obtained from a classical phase-space distribution. We explore, for both the classical and quantum case, the relationship of the range of the current with the range of the momentum distribution function. We further show that for windowed type distributions something interesting happens, in that it is possible to choose a window which yields a classical type result, namely, such that the total current for a multi-part wave function is a weighted average of the currents corresponding to each part. We give the conditions for when this is the case.