Skip to content

Lecture 5.1: The Wigner Quasi-Probability Distribution

Expected prior knowledge

Before the start of this lecture, you should be able to:

  1. Analyze the properties of the coherent state wavefunction
  2. Recall that describes a probability distribution

Learning goals

After this lecture you will be able to:

  1. Describe the main properties of the Wigner function
  2. Describe the relation between the Wigner function and the probability distribution of a quantum state

In the last lecture we discussed some properties of quantum fluctuations of the ground state of the harmonic oscillator, and took a look at an interesting state of the quantum harmonic oscillator known as the coherent state.

We will now inspect the coherent state in more detail, but first we will study a different way of representing the wavefunction called the Wigner quasiprobability distribution, or for short, the Wigner function.

  1. For a single particle in one dimension in quantum state with wavefunction , the Wigner function is defined as where is just the 1D scalar momentum, not the momentum operator.

  2. We can also form with , the Fourier transform of :

How should we interpret the Wigner function?

It must be something like a probability distribution, as the name suggests.

  1. If we integrate over all , we get : which is a probability distribution.

  2. Similarly, if we integrate over all , we get .

So it's close, but there are two problems with calling a probability distribution:

  • In quantum mechanics, we can only measure one thing at a time, and after measuring our wave function, it collapses, so cannot be the probability of measuring both and , which are non-commuting observables

  • for some interesting quantum states, can be negative, and since probabilities cannot be negative, we must call it a quasiprobability distribution instead.

Note that although can be negative, it must be constructed such that its integrals over and are positive, since these do indeed represent probability distributions, as shown above.

Also note that the Wigner function is real-valued, though it may not be obvious from the formula above.

To make this more concrete, let's calculate the Wigner function of the ground state of the harmonic oscillator:

  1. Defining , we can write

  2. Plugging into the Wigner function, we get

  3. After working out the exponents, we get which is just the Fourier transform of a Gaussian, which is still a Gaussian.

  4. Thus, we have This is a two-dimensional Gaussian "disk" centered around ;

In the next two sections of this lecture, we will explore the Wigner functions of different quantum states using a Python library called QuTiP.

The homework for this lecture is at the bottom of the last section ("Wigner Functions of Mixed States").