Phase space and the quantum Liouville theorem

[skynelson]

I would like to understand phase space better, spec. in relation to the quantum Liouville theorem. Can anyone point me to a decent online resource? I am most interested in conceptual understanding to begin with.

Liouville's theorem says that if you follow a point in phase space, the number of points surrounding that point will always be the same. Or, equivalently, "The distribution function is constant along any trajectory in phase space."

In the case of a quantum density matrix, what does the phase space represent? The various possible states and their probabilities?

What, then, is an intuitive interpretation of the "points in phase space", and their "spacing"?

 

[xepma]

The quantum analog of the distribution function on the classical phase space is called the Wigner distribution function. It's basically a Fourier transform-like representaton of the density matrix. It reduces to the classical distribution function you mentioned in the classical limit. But unlike the classical counterpart the Wigner distribution can take on negative values as well.

The "equation of motion" this object satisfies is the quantum version of the Liouville equation.

You cannot really talk about a point in phase space in the quantum sense. The phase space is classically spanned by coordinates x and momentum p. But in quantum mechanics these variables are turned into operators. The fundamental reason for this is the assumption of non-commutativity: [x,p] = ih. This relation can only be satisfied by assuming the variables are now operators.

But a more important consequence is that we cannot think of x and p as spanning an ordinary phase space. The reason is that the non-commutative property does not allow us to simultaneously specify both momentum and coordinate with infinite precision. That means we cannot talk about coordinates of a space in the usual sense. Rather, we are dealing with what is called a non-commutative geometry: a space in which coordinates in different directions fail to commute. This goes beyond my knowledge though, but it should be clear that the picture of an ordinary phase space breaks down because of this. It is replaced by quantum states and density matrices.