Suppose that A=A(P), i.e., a function of the momentum operator. Then, the trace can still be evaluated in the coordinate basis:
However, A(P) acting to the left does not act on an eigenvector.
Let us insert a coordinate space identity 
between A and 
:
Now, we see that the expectation value can be obtained by evaluating all the coordinate space matrix elements of the operator and all the coordinate space matrix elements of the density matrix.
A particularly useful form for the expectation value can be obtained if a momentum space identity is inserted:
Now, we see that A(P) acts on an eigenstate (at the price of introducing another integral). Thus, we have
Using the fact that 
, we find
that
In the above expression, we introduce the change of variables
Then
Define a distribution function
Then, the expectation value can be written as
which looks just like a classical phase space average using
the ``phase space'' distribution function 
.
The distribution function is known as the
Wigner density matrix and it has many interesting features.
For one thing, its classical limit is
which is the true classical phase space distribution function. There are various examples, in which the exact Wigner distribution function is the classical phase space distribution function, in particularly for quadratic Hamiltonians. Despite its compelling appearance, the evaluation of expectation values of functions of momentum are considerably more difficult than functions of position, due to the fact that the entire density matrix is required. However, there are a few quantities of interest, that are functions of momentum, that can be evaluated without resorting to the entire density matrix. These are thermodynamic quantities which will be discussed in the next section.