TY - JOUR

T1 - On the distribution of the wave function for systems in thermal equilibrium

AU - Goldstein, Sheldon

AU - Lebowitz, Joel L.

AU - Tumulka, Roderich

AU - Zanghì, Nino

N1 - Funding Information:
The work of S. Goldstein was supported in part by NSF Grant DMS-0504504, and that of J. Lebowitz by NSF Grant DMR 01-279-26 and AFOSR Grant AF 49620-01-1-0154. The work of R. Tumulka was supported by INFN and by the European Commission through its 6th Framework Programme “Structuring the European Research Area” and the contract Nr. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action. The work of N. Zanghìwas supported by INFN.

PY - 2006/12

Y1 - 2006/12

N2 - For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution μ of its wave function, a normalized vector belonging to its Hilbert space. While ρ itself does not determine a unique μ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which μ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in, denoted GAP(ρ). We do this using a suitable projection of the Gaussian measure on with covariance ρ. We establish some nice properties of GAP(ρ) 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 ρβ = (1/Z) exp (-β H). GAP(ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on are often used.

AB - For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution μ of its wave function, a normalized vector belonging to its Hilbert space. While ρ itself does not determine a unique μ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which μ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in, denoted GAP(ρ). We do this using a suitable projection of the Gaussian measure on with covariance ρ. We establish some nice properties of GAP(ρ) 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 ρβ = (1/Z) exp (-β H). GAP(ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on are often used.

KW - Canonical ensemble in quantum theory

KW - Density matrices

KW - Gaussian measures

KW - Probability measures on Hilbert space

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U2 - 10.1007/s10955-006-9210-z

DO - 10.1007/s10955-006-9210-z

M3 - Article

AN - SCOPUS:33845890805

SN - 0022-4715

VL - 125

SP - 1193

EP - 1221

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 5-6

ER -