7 research outputs found
On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems
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.Comment: 19 pages LaTeX, no figure
Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment
A quantum system (with Hilbert space ) entangled with its
environment (with Hilbert space ) is usually not attributed a
wave function but only a reduced density matrix . Nevertheless, there
is a precise way of attributing to it a random wave function , called
its conditional wave function, whose probability distribution depends
on the entangled wave function in
the Hilbert space of system and environment together. It also depends on a
choice of orthonormal basis of but in relevant cases, as we
show, not very much. We prove several universality (or typicality) results
about , e.g., that if the environment is sufficiently large then for
every orthonormal basis of , most entangled states with
given reduced density matrix are such that is close to one of
the so-called GAP (Gaussian adjusted projected) measures, . We
also show that, for most entangled states from a microcanonical subspace
(spanned by the eigenvectors of the Hamiltonian with energies in a narrow
interval ) and most orthonormal bases of ,
is close to with the
normalized projection to the microcanonical subspace. In particular, if the
coupling between the system and the environment is weak, then is close
to with the canonical density matrix on
at inverse temperature . This provides the
mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006),
http://arxiv.org/abs/quant-ph/0309021] that measures describe the thermal
equilibrium distribution of the wave function.Comment: 27 pages LaTeX, no figures; v2 major revision with simpler proof
Normal Typicality and von Neumann’s Quantum Ergodic Theorem
We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e., the statement that, for typical large systems, every initial wave function ψ0 from an energy shell is “normal”: it evolves in such a way that |ψt〉〈ψt | is, for most t, macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof