On the distribution of the wave function for systems in thermal equilibrium
On the distribution of the wave function for systems in thermal equilibrium
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DOI：
10.1007/s10955-006-9210-z
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发表时间：
2006-12-01
影响因子：
1.6
通讯作者：
Zanghi, Nino
中科院分区：
文献类型：
--
作者：
Goldstein, Sheldon;Lebowitz, Joel L.;Zanghi, Nino
对于一个量子系统，一个不纯的密度矩阵rho可以通过平均从它的波函数的分布μ中产生，一个属于它的希尔伯特空间H的归一化向量。虽然ρ本身并不确定一个唯一的μ，但其他的事实，如系统已经达到热平衡，可能。因此，提出这样一个问题并不是没有道理的：如果有的话，哪个μ对应于给定的热力学系综？为了回答这个问题，我们构造，对于任何给定的密度矩阵ρ，在H中的单位球面上的自然测度，记为GAP（ρ）。我们使用高斯测度在H上的适当投影与协方差ρ来做到这一点。我们建立了一些很好的性质的差距（ρ），并表明，这种措施自然出现时，考虑宏观系统。特别是，我们认为，它是最合适的选择系统的热平衡，描述的正则系综密度矩阵ρ（β）=（1/Z）exp（-β H）。间隙（ρ）也可能与量子混沌和开放量子系统的随机演化有关，其中经常使用H上的分布。
For a quantum system, a density matrix rho that is not pure can arise, via averaging, from a distribution mu of its wave function, a normalized vector belonging to its Hilbert space H. While rho itself does not determine a unique mu, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which mu, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix rho, a natural measure on the unit sphere in H, denoted GAP(rho). We do this using a suitable projection of the Gaussian measure on H with covariance rho. We establish some nice properties of GAP(rho) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix rho(beta)=(1/Z) exp (-beta H). GAP(rho) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on H are often used.