*[Follow-up post: In what sense is the Wigner function a quasiprobability distribution?]*

I’ve never liked how people introduce the Wigner function (aka the Wigner quasi-probability distribution). Usually, they just write down a definition like

(1)

and say that it’s the “closest phase-space representation” of a quantum state. One immediately wonders: What’s with the weird factor of , and what the heck is ? Usually, the only justification given for the probability interpretation is that integrating over one of the variables recovers the probability distribution for the other (if it were measured):

(2)

where is just the density matrix in the momentum basis. But of course, that doesn’t really tell us why we should think of , as having anything to do with the (rough) value of conditional on a (rough) value of .

Well now I have a much better idea of what the Wigner function actually is and how to interpret it. We start by writing it down in sane variables (and suppress ):

(3)

So the first step in the interpretation is to consider the function

(4)

which appears in the integrand. This is just the (position-space) density matrix in rotated coordinates and . There is a strong sense in which the off-diagonal terms of the density matrix represent the quantum coherence of the state between different positions, so indexes how far this coherence extends; large values of indicate large spatial coherence. On the other hand, indexes how far down the diagonal of the density matrix we move; it’s the average position of the two points between which the off-diagonal terms of the density matrix measures coherence. (See the figure below.)… [continue reading]