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Ergodicity of quantum mechanical systems
1.J. von Neumann, Z. Phys. 57, 30 (1929).
2.D. ter Haar, Elements of Statistical Mechanics (Holt, Rinehart and Winston, New York, 1961), Appendix I and references cited therein.
3.K. Matsuno, J. Math. Phys. 16, 604 (1975).
4.Here, we understand that there exist at least two kinds of probability, each of which may follow the axioms of probability. One is the logical probability of expectation which is specific to a subjective observer. The other is the statistically inductive probability as relative frequency which is frequently used in ensemble theory of stochastic process, although it would not be certain whether the probability certainly converges as absolute frequency increases. The equal a priori probability of expectation should be distinguished from the principle of equal a priori probabilities (cf. Ref. 2 and Sec. 4 of this paper). The a priori probability of expectation is specific to an observer. However, a priori probabilities used in the context of the principle of equal a priori probabilities are not quite clear with respect to whether they refer to the logical probabilities of expectation or to the statistical probabilities as relative frequencies. In order to avoid unnecessary complexities, we use the term the principle of (a priori) equal weight in Ref. 3 in place of the principle of equal a priori probabilities.
5.L. D. Landau and E. M. Llfshltz, Statistical Physics (Pergamon, London, 1969).
6.The time evolution of the density matrix is free from the contraction of wavefunctlons. Hence, one is not sure whether any observable averaged over a statistical ensemble could certainly be physically measurable, since the contraction of wavefunctions, which is inevitable for real measurements, is absent in this scheme of quantum statistics.
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