In general, the many-body wave function 
is far too
large to calculate for a macroscopic system. If we wish to represent it on
a grid with just 10 points along each coordinate direction, then for
, we would need
total points, which is clearly enormous.
We wish, therefore, to use the concept of ensembles in order to
express expectation values of observables 
without requiring
direct computation of the wavefunction. Let us, therefore, introduce
an ensemble of systems, with a total of Z members, and each having
a state vector 
, 
. Furthermore, introduce an
orthonormal set of vectors 
( 
)
and expand the state vector for each member of the ensemble in this orthonormal
set:
The expectation value of an observable, averaged over the ensemble of systems is given by the average of the expectation value of the observable computed with respect to each member of the ensemble:
Substituting in the expansion for , we obtain
Let us define a matrix
and a similar matrix
Thus, 
is a sum over the ensemble members of a product
of expansion coefficients, while 
is an average
over the ensemble of this product. Also, let 
.
Then, the expectation value can be written as follows:
where 
and A represent the matrices with elements and
in the basis of vectors 
. The matrix
is known as the density matrix. There is an abstract operator
corresponding to this matrix that is basis-independent. It can be seen that
the operator
and similarly
have matrix elements when evaluated in the basis set of
vectors .
Note that is a hermitian operator
so that its eigenvectors form a complete orthonormal set of vectors
that span the Hilbert space. If 
and 
represent
the eigenvalues and eigenvectors of the operator 
, respectively,
then several important properties they must satisfy can be deduced.
Firstly, let A be the identity operator I. Then, since 
,
it follows that
Thus, the eigenvalues of must sum to 1. Next, let A
be a projector onto an eigenstate of , 
.
Then
But, since can be expressed as
and the trace, being basis set independent, can be therefore be evaluated
in the basis of eigenvectors of , the expectation value becomes
However,
Thus, 
. Combining these two results, we see that,
since 
and , 
, so that
satisfy the properties of probabilities.
With this in mind, we can develop a physical meaning for the density matrix.
Let us now consider the expectation value of a projector 
onto one of the eigenstates of the operator A. The expectation value of
this operator is given by
But 
is just probability that a measurement of the
operator A in the 
th member of the ensemble will yield
the result 
. Thus,
or the expectation value of 
is just the ensemble averaged
probability of obtaining the value in each member of the ensemble.
However, note that the expectation value of can
also be written as
Equating the two expressions gives
The interpretation of this equation is that the ensemble averaged probability
of obtaining the value if A is measured is equal to the
probability of obtaining the value in a measurement of A
if the state of the system under consideration were the state ,
weighted by the average probability that the system in the ensemble is in that state.
Therefore, the density operator (or ) plays the
same role in quantum systems that the phase space distribution
function 
plays in classical systems.