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This work describes the statistics for the occupation numbers of quantumlevels in a large isolated quantum system, where all possible superpositions ofeigenstates are allowed, provided all these superpositions have the same fixedenergy. Such a condition is not equivalent to the conventional micro-canonicalcondition, because the latter limits the participating eigenstates to a verynarrow energy window. The statistics is obtained analytically for both theentire system and its small subsystem. In a significant departure from theBoltzmann-Gibbs statistics, the average occupation numbers of quantum statesexhibit in the present case weak algebraic dependence on energy. In themacroscopic limit, this dependence is routinely accompanied by the condensationinto the lowest energy quantum state. This work contains initial numericaltests of the above statistics for finite systems, and also reports thefollowing numerical finding: When the basis states of large but finite randommatrix Hamiltonians are expanded in terms of eigenstates, the participation ofeigenstates in such an expansion obeys the newly obtained statistics. The abovestatistics might be observable in small quantum systems, but for themacroscopic systems, it rather reenforces doubts about self-sufficiency ofnon-relativistic quantum mechanics for justifying the Boltzmann-Gibbsequilibrium.

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