On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems

Details On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems

2009-11-09

arxiv.org/abs/0911.1724v1

Phys. Rev. E 81: 011109 (2010)

Physics

Quantum Physics

19 pages LaTeX, no figures

Scientific paper

10.1103/PhysRevE.81.011109

We consider an isolated, macroscopic quantum system. Let H be a micro-canonical "energy shell," i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E + delta E. The thermal equilibrium macro-state at energy E corresponds to a subspace H_{eq} of H such that dim H_{eq}/dim H is close to 1. We say that a system with state vector psi in H is in thermal equilibrium if psi is "close" to H_{eq}. We show that for "typical" Hamiltonians with given eigenvalues, all initial state vectors psi_0 evolve in such a way that psi_t is in thermal equilibrium for most times t. This result is closely related to von Neumann's quantum ergodic theorem of 1929.

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