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
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
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.