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Molecular Physics

S-matrix theory and the density matrix formalism

M. Roberts, W. E. Hagston

Published Aug 1, 1979• 2 citations
Publisher

Abstract

Here we extend the previous treatment [1] to a discussion of open systems, and are led to a derivation of the generalized Pauli equation. Consideration of these results leads to the proposal of a more general equation involving the total density matrix operator. The weak coupling approximation of the latter leads once again to the generalized Pauli equation.

DOI
10.1080/00268977900101881

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Here we extend the previous treatment [1] to a discussion of open systems, and are led to a derivation of the generalized Pauli equation. Consideration of these results leads to the proposal of a more general equation involving the total density matrix operator. The weak coupling approximation of the latter leads once again to the generalized Pauli equation.
Treatment of the predictive aspect of statistical mechanics as a form of statistical inference is extended to the density-matrix formalism and applied to a discussion of the relation between irreversibility and information loss. A principle of "statistical complementarity" is pointed out, according to which the empirically verifiable probabilities of statistical mechanics necessarily correspond to incomplete predictions. A preliminary discussion is given of the second law of thermodynamics and of a certain class of irreversible processes, in an approximation equivalent to that of the semiclassical theory of radiation.It is shown that a density matrix does not in general contain all the information about a system that is relevant for predicting its behavior. In the case of a system perturbed by random fluctuating fields, the density matrix cannot satisfy any differential equation because $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}(t)$ does not depend only on $\ensuremath{\rho}(t)$, but also on past conditions The rigorous theory involves stochastic equations in the type $\ensuremath{\rho}(t)=\mathcal{G}(t, 0)\ensuremath{\rho}(0)$, where the operator G\mathcal{G}G is a functional of conditions during the entire interval ($0\ensuremath{\rightarrow}t$). Therefore a general theory of irreversible processes cannot be based on differential rate equations corresponding to time-proportional transition probabilities. However, such equations often represent useful approximations.
Previous work extending the Kohn–Sham theory to excited states was based on replacing the study of the ground-state energy as a functional of the ground-state density by a study of an ensemble average of the Hamiltonian as a functional of the corresponding average density. We suggest and develop an alternative to this description of excited states that utilizes the matrix of the density operator taken between any two states of the included space. Such an approach provides more detailed information about the states included, for example transition probabilities between discrete states of local one-body operators. The new theory is also based on a variational principle for the trace of the Hamiltonian over the space of states that we wish to describe, viewed, however, as a functional of the associated array of matrix elements of the density. This finds expression in a matrix version of the Kohn–Sham theory. To illustrate the formalism, we study a suitably defined weak-coupling limit, which is our equivalent of the linear response approximation. On this basis, we derive an eigenvalue equation that has the same form as an equation derived directly from the time-dependent Kohn–Sham equation and applied recently with considerable success to molecular excitations. We provide an independent proof, within the defined approximations, that the eigenvalues can be interpreted as true excitation energies.