The time dependent wave equation for the macroscopic pair wave function in superconductors is derived in a macroscopic manner from the assumption of off diagonal long range order and the neglect of long range three-particle correlations. It is shown that the equation is satisfied in equilibrium by the conventional microscopic theory.
Abstract A qualitative description of the modern microscopic theory of superconductivity is given, concentrating on the basic physical ideas involved.
Zwanzig's relaxation equation for a system in contact with a heat bath is further developed in a general way by fully exploiting the limit of an infinite bath. The relaxation operator is expressed as a true bath correlation average. The classical and quantum versions are exhibited by means of Prigogine's matrix notation and the relaxation operator shown to be a renormalized version of the Prigogine type of relaxation operator. The exact relaxation equation of a classical Brownian oscillator is derived in the long time limit as an illustration of the technique.
The present paper derives conditions on which the Boltzmann equations for electron-phonon systems (two coupled equations) can be derived without the use of the objectionable random phase approximation. The treatment uses von Neumann's density matrix based on the Hamiltonian introduced some time ago by one of us in connection with superconductivity. Formally exact expressions for the rate of change of coarse-grained one-particle distribution functions are derived. They become identical with the collision terms in the Boltzmann equations if (i) the influence of an external field on them is negligible, (iia) particle correlation is absent at an initial time, and (iib) the relevant series in terms of the interaction parameter converges and is at all times dominated by a certain term. (iib) excludes the possibility of extensive correlations as they occur in superconductivity.
The general macroscopic description of the electromagnetic behaviour of superconductors and normal conductors is developed and dispersion relations for the frequency dependent material functions are rigorously established. From these dispersion relations expressions similar to the usual sum rules of electromagnetic theory are derived. Using these, Ferrell's proof that the Meissner effect is a consequence of the energy gap is shown to be unsound. Using the complex surface impedance it is shown macroscopically that as a consequence of the Meissner effect the area under the absorptivity curve for a superconductor is less than for the normal state.
