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The density matrix and density operator

In general, the many-body wave function tex2html_wrap_inline485 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 tex2html_wrap_inline487 , 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 tex2html_wrap_inline489 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 tex2html_wrap_inline493 , tex2html_wrap_inline495 . Furthermore, introduce an orthonormal set of vectors tex2html_wrap_inline497 ( tex2html_wrap_inline499 ) 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 tex2html_wrap_inline493 , we obtain


Let us define a matrix


and a similar matrix


Thus, tex2html_wrap_inline503 is a sum over the ensemble members of a product of expansion coefficients, while tex2html_wrap_inline505 is an average over the ensemble of this product. Also, let tex2html_wrap_inline507 . Then, the expectation value can be written as follows:


where tex2html_wrap_inline509 and A represent the matrices with elements tex2html_wrap_inline503 and tex2html_wrap_inline515 in the basis of vectors tex2html_wrap_inline517 . The matrix tex2html_wrap_inline503 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 tex2html_wrap_inline503 when evaluated in the basis set of vectors tex2html_wrap_inline517 .


Note that tex2html_wrap_inline509 is a hermitian operator


so that its eigenvectors form a complete orthonormal set of vectors that span the Hilbert space. If tex2html_wrap_inline527 and tex2html_wrap_inline529 represent the eigenvalues and eigenvectors of the operator tex2html_wrap_inline531 , respectively, then several important properties they must satisfy can be deduced.

Firstly, let A be the identity operator I. Then, since tex2html_wrap_inline537 , it follows that


Thus, the eigenvalues of tex2html_wrap_inline531 must sum to 1. Next, let A be a projector onto an eigenstate of tex2html_wrap_inline531 , tex2html_wrap_inline545 . Then


But, since tex2html_wrap_inline531 can be expressed as


and the trace, being basis set independent, can be therefore be evaluated in the basis of eigenvectors of tex2html_wrap_inline531 , the expectation value becomes




Thus, tex2html_wrap_inline551 . Combining these two results, we see that, since tex2html_wrap_inline553 and tex2html_wrap_inline551 , tex2html_wrap_inline557 , so that tex2html_wrap_inline527 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 tex2html_wrap_inline561 onto one of the eigenstates of the operator A. The expectation value of this operator is given by


But tex2html_wrap_inline565 is just probability that a measurement of the operator A in the tex2html_wrap_inline569 th member of the ensemble will yield the result tex2html_wrap_inline571 . Thus,


or the expectation value of tex2html_wrap_inline573 is just the ensemble averaged probability of obtaining the value tex2html_wrap_inline571 in each member of the ensemble. However, note that the expectation value of tex2html_wrap_inline573 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 tex2html_wrap_inline571 if A is measured is equal to the probability of obtaining the value tex2html_wrap_inline571 in a measurement of A if the state of the system under consideration were the state tex2html_wrap_inline529 , weighted by the average probability tex2html_wrap_inline527 that the system in the ensemble is in that state. Therefore, the density operator tex2html_wrap_inline509 (or tex2html_wrap_inline531 ) plays the same role in quantum systems that the phase space distribution function tex2html_wrap_inline595 plays in classical systems.

next up previous
Next: Time evolution of the Up: Principles of quantum statistical Previous: Principles of quantum statistical

Mark Tuckerman
Tue May 9 19:40:24 EDT 2000