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

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.

20 pages, 3 figures

References in corpus (8)

• Thermalization and its mechanism for generic isolated quantum systems
• The foundations of statistical mechanics from entanglement: Individual states vs. averages
• Canonical Typicality
• Foundation of Statistical Mechanics under experimentally realistic conditions
• Universal Long-time Behavior of Nuclear Spin Decays in a Solid
• Quantum phase transitions without thermodynamic limits
• Solvable model of quantum microcanonical states
• A generalized quantum microcanonical ensemble

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