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Progress of Theoretical Physics

The Expectation Value of a Many-Body Hamiltonian in the Quasi-Chemical Equilibrium Theory

James McKenna, John M. Blatt

Published Mar 1, 1962• 2 citations
Publisher

Abstract

For the purpose of studying the Meissner effect in superconductivity, a Hamiltonian is considered which includes both Coulomb interactions and electron-phonon interactions. The expectation value of this Hamiltonian is evaluated in the quasi-chemical equilibrium approximation to statistical mechanics. A compact expression is obtained for the expectation value of the Hamiltonian which, it is hoped, will be useful in a variational treatment of the Meissner effect.

DOI
10.1143/ptp.27.511

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For the purpose of studying the Meissner effect in superconductivity, a Hamiltonian is considered which includes both Coulomb interactions and electron-phonon interactions. The expectation value of this Hamiltonian is evaluated in the quasi-chemical equilibrium approximation to statistical mechanics. A compact expression is obtained for the expectation value of the Hamiltonian which, it is hoped, will be useful in a variational treatment of the Meissner effect.
A new variational method for studying the equilibrium states of an interacting particles system has been proposed. The statistical description of the system is realized by means of a density matrix. This method is used for description of conduction electrons in metals. An integral equation for the electron distribution function over wave vectors has been obtained. The solutions of this equation have been found for those cases where the single-particle Hamiltonian and the electron interaction Hamiltonian can be approximated by a quite simple expression. It is shown that the distribution function at temperatures below the critical value possesses previously unknown features which allow to explain the superconductivity of metals and presence of a gap in the energy spectrum of superconducting electrons.
BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).
The equilibrium statistical mechanics of the Bardeen-Cooper-Schrieffer model of superconductivity, as well as that of a wide class of similar models, can be evaluated exactly by the ``thermodynamically equivalent Hamiltonian'' method of Bogoliubov, Zubarev, Tserkovnikov, and Wentzel. It has been pointed out by Wentzel that this method can be extended to certain weakly nonequilibrium situations. It is shown here that the method allows an exact evaluation of the nonequilibrium statistical mechanics of the following situations: (a) temporal evolution of the statistical expectation value of an observable O(r) whose initial deviation from equilibrium is spatially localized, but not necessarily small; (b) temporal evolution of the statistical expectation value of an observable O due to a perturbation V which is spatially localized, but not necessarily small.