能量固定且本征态参与不受限制的孤立量子系统的典型态
Typical state of an isolated quantum system with fixed energy and unrestricted participation of eigenstates
摘要：
本文描述了一个孤立的大量子系统中量子能级占据数的统计，在这个系统中，所有可能的本征态叠加都是允许的，只要这些叠加态具有相同的固定能量。这样的条件不等同于传统的微正则条件，因为后者将参与的本征态限制在非常窄的能量窗口。整个系统和它的小的子系统的统计分析。与玻尔兹曼-吉布斯统计显著不同的是，量子态的平均占据数在本例中表现出对能量的弱代数依赖性。在宏观极限下，这种依赖性通常伴随着向最低能量量子态的凝聚。这项工作包含上述统计有限系统的初始数值测试，并报告了以下数值发现：当大而有限的随机矩阵哈密顿的基态扩展的本征态，本征态的参与，这样的扩展服从新获得的统计。上述统计在小量子系统中可能是可观察到的，但对于宏观系统，它反而加强了对非相对论性量子力学证明玻尔兹曼-吉布斯平衡的自足性的怀疑。
英文摘要：
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.
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