For the physical situation which occupies our concern, we can most easily denote the density operator in matrix form:
Each k corresponds to a different possible atom in our system. Multiplying out the k vectors provides a density matrix for the kth atom:
Notice that the elements on the diagonal of the density matrix are real numbers which correspond to the probability of being in the upper or lower states. The sum of these two diagonal elements should be unity by the normalization criterion which is a general property of the density matrix, . The off--diagonal elements, on the other hand, are complex numbers whose meaning is related to the phase of the configuration's wavefunction.
Uncovering the semiclassical equations in this picture is a relatively simple affair. Find the total hamiltonian for the system and apply eq:T10. However, there is another way to solve this problem. We have already uncovered the time dependence of the probability amplitudes for the upper and lower states. The density matrix is obviously related to these amplitudes. Therefore, one must merely find how the time dependence of the elements in the density matrix is related to the probability amplitudes.
This dependence is fairly easy to uncover by differentiating the kth configuration's density matrix with respect to time
The derivative must be treated like a scalar, linear operator. It will operate on each element of the matrix. Notice that the product rule will be used since each element is the product of two time dependent functions
eq:T8 and eq:T9 provide us with both and ; however, we also require the conjugates of these terms which are
Substituting eq:T8, eq:T9, eq:T12, and eq:T13 into eq:T11 generates
By collecting all the terms together and applying the definitions of the density matrix elements, each equation can be simplified
This set of four equations defines the semiclassical theory of atom--field interactions in the density matrix picture. We are able to obtain the same information that the Schrödinger picture offered via and ; however, information about the phase of the wavefunction can be found in the off--diagonal elements of the density matrix. Note that and are conjugates of each other and that the diagonal elements are real. The density matrix is hermitian, and this fact will be used again and again to simplify the numerical algorithms presented in this text. Here, for example, only one diagonal element and a single off--diagonal element are required to completely specify this matrix.