4 - Quasiprobability distributions

Summary

Wigner representation

In classical optics (Born and Wolf, 1999), the state of the electromagnetic oscillator is perfectly described by the statistics of the classical amplitude α. The amplitude may be completely fixed (then the field is coherent), or αmay fluctuate (then the field is partially coherent or incoherent). In classical optics as well as in classical mechanics, we can characterize the statistics of the complex amplitude?or, equivalently, the statistics of the components position qand momentum pintroducing a phase space distribution W(q, p). (As explained in Section 3.1, the real and the imaginary part of the complex amplitude αcan be regarded as the position and the momentum of the electromagnetic oscillator.) The distribution W(q, p) quantifies the probability of finding a particular pair of qand pvalues in their simultaneous measurement. Knowing the phase space probability distribution, all statistical quantities of the electromagnetic oscillator can be predicted by calculation. In this sense the phase space distribution describes the state in classical physics.

All this is much more subtle in quantum mechanics. First of all, Heisenberg's uncertainty principle prevents us from observing position momentum simultaneously andprecisely. So it seems there is no point in thinking about quantum phase space. But wait! In quantum mechanics we cannot directly observe quantum states either. Nevertheless, we are legitimately entitled to use the concept of states as if they were existing entities (whatever they are). We use their properties to predict the statistics of observations.