“Given a wavefunction , the Wigner function is a function over classical phase space, defined by (5.77) We want to think of this as something akin to a probability distribution over phase space. At first glance that seems unlikely because, as we’ve seen, there is a difference between quantum states whose properties are undetermined and classical probability that can be ascribed to ignorance. This is reflected in the fact that and so we can’t ascribe simultaneous values to both observables. And, indeed, it will turn out that it’s not possible to interpret as a classical probability distribution. Nonetheless, it gets close. Let’s look at some properties of the Wigner function. First, it is real. (This follows by taking the complex conjugate and changing variables to .) Second, if we integrate over momentum, and use the fact that , we have (5.78) But that’s rather nice: marginalising over momentum gives us , which we know is the probability distribution over position. Moreover, if we have a normalised wavefunction then we know that .”