Typical state of an isolated quantum system with fixed energy and unrestricted participation of eigenstates

TL;DR

This paper derives and tests a new statistical description for occupation numbers in isolated quantum systems with fixed energy, revealing deviations from classical thermodynamics and potential implications for quantum mechanics foundations.

Contribution

It introduces a novel statistical framework for quantum superpositions at fixed energy, differing from micro-canonical ensemble, with analytical results and numerical validation.

Findings

Occupation numbers show weak algebraic energy dependence.

Condensation into the lowest energy state occurs in the macroscopic limit.

Eigenstate participation in random matrix Hamiltonians follows the new statistics.

Abstract

This work describes the statistics for the occupation numbers of quantum levels in a large isolated quantum system, where all possible superpositions of eigenstates are allowed, provided all these superpositions have the same fixed energy. Such a condition is not equivalent to the conventional micro-canonical condition, because the latter limits the participating eigenstates to a very narrow energy window. The statistics is obtained analytically for both the entire system and its small subsystem. In a significant departure from the Boltzmann-Gibbs statistics, the average occupation numbers of quantum states exhibit in the present case weak algebraic dependence on energy. In the macroscopic limit, this dependence is routinely accompanied by the condensation into the lowest energy quantum state. This work contains initial numerical tests of the above statistics for finite systems, and also reports the following numerical finding: When the basis states of large but finite random matrix Hamiltonians are expanded in terms of eigenstates, the participation of eigenstates in such an expansion obeys the newly obtained statistics. The above statistics might be observable in small quantum systems, but for the macroscopic systems, it rather reenforces doubts about self-sufficiency of non-relativistic quantum mechanics for justifying the Boltzmann-Gibbs equilibrium.