Microcanonical distributions for quantum systems

TL;DR

This paper proposes a generalized microcanonical ensemble in quantum mechanics allowing superpositions with fixed energy expectation, deriving a density of states formula and exploring its implications for quantum systems.

Contribution

It introduces a weakened microcanonical postulate permitting superpositions, providing a closed-form density of states for nondegenerate spectra, and analyzing its behavior in large systems.

Findings

Derived a closed-form density of states for quantum systems.

Showed convergence of the density of states distribution as energy levels increase.

Explored the relation between energy and temperature in the new framework.

Abstract

The standard assumption for the equilibrium microcanonical state in quantum mechanics, that the system must be in one of the energy eigenstates, is weakened so as to allow superpositions of states. The weakened form of the microcanonical postulate thus asserts that all quantum states giving rise to the same energy expectation value must be realised with equal probability. The consequences that follow from this assertion are investigated. In particular, a closed-form expression for the density of states associated with any system having a nondegenerate energy spectrum is obtained. The result is applied to a variety of examples, for which the behaviour of the state density, as well as the relation between energy and temperature, are determined. Numerical studies indicate that the density of states converges to a distribution when the number of energy levels approaches infinity.