Solvable model of quantum microcanonical states

Abstract

This letter examines the consequences of a recently proposed modification of the postulate of equal {\it a priori} probability in quantum statistical mechanics. This modification, called the {\it quantum microcanonical postulate} (QMP), asserts that for a system in microcanonical equilibrium all pure quantum states having the same energy expectation value are realised with equal probability. A simple model of a quantum system that obeys the QMP and that has a nondegenerate spectrum with equally spaced energy eigenvalues is studied. This model admits a closed-form expression for the density of states in terms of the energy eigenvalues. It is shown that in the limit as the number of energy levels approaches infinity, the expression for the density of states converges to a $δ$ function centred at the intermediate value $(E_{\rm max}+E_{\rm min})/ 2$ of the energy. Determining this limit requires an elaborate asymptotic study of an infinite sum whose terms alternate in sign.