Goldstein Sheldon, Lebowitz Joel L, Tumulka Roderich, Zanghì Nino
Department of Mathematics, Hill Center, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA.
Phys Rev Lett. 2006 Feb 10;96(5):050403. doi: 10.1103/PhysRevLett.96.050403. Epub 2006 Feb 8.
It is well known that a system weakly coupled to a heat bath is described by the canonical ensemble when the composite S + B is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true for both classical distributions on the phase space and quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands, and justifies remarks made by Schrödinger in 1952.
众所周知，当复合系统S + B由对应于合适能量壳层的微正则系综描述时，与热库弱耦合的系统由正则系综描述。这对于相空间上的经典分布和量子密度矩阵都是成立的。在这里我们表明，对于量子系统有一个更强的结论成立。即使复合系统的状态对应于单个波函数而不是混合态，对于微正则系综所包含的能量区间对应的子空间中的绝大多数波函数，系统的约化密度矩阵也是正则的。这澄清、扩展并证明了薛定谔在1952年所做的论述。
