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The Density Matrix

The derivation of the equations of motion for this model has used the Schrödinger picture exclusively. There is a better way of examining this problem which allows the addition of a number of new physical effects. This new picture is called the density matrix picture. We could have derived the basic equations of motion in this formalism; however, it is difficult to improve on the elegant derivation which the Schrödinger picture offers in this case. Then one might ask why we appeal to this formalism now. The answer lies in the underlying statistical properties of the density matrix. A basic discussion of this formalism can be found in [3].

One can best utilize the Schrödinger picture when one knows the complete wavefunction for the system. If we were interested in examining only one atom interacting with a classical radiation field, then we need not go any further. Suppose, on the other hand, that we wish to examine how a large number of atoms interact with a field. In this case, the Schrödinger picture becomes unwieldy because we would have to solve for the probability amplitudes of each atom. This task is beyond the capabilities of any supercomputer for the number of atoms in any macroscopic sample. The density matrix is our saviour, for it does not require a complete wavefunction for a system. Using this formalism, we can treat a huge number of atoms interacting with a field via statistical means. It is this capability which makes the density matrix appealing in this situation.

The density operator is defined as:

where is the probability of the configuration in some ensemble. The density operator can be written in the form of a matrix, the density matrix. The terminology is used interchangeably in the literature, for in Heisenberg's matrix mechanics a matrix is an operator.

Before proceeding, a few familiar ideas will be defined in this new formalism. We could have discovered the equations of motion for the system at hand by jumping directly into this picture via


which is equivalent to the time--dependent Schrödinger equation. Another common definition that will be needed is the expectation value

where denotes the trace of the matrix .

next up previous contents index
Next: Density Matrix Derivation Up: Theory Previous: Assumptions and Approximations

Andy Antonelli
Wed May 17 14:34:24 EDT 1995