We consider the thermal equilibrium distribution at inverse temperature β, or canonical ensemble, of the wave function Ψ of a quantum system. Since L 2 spaces contain more nondifferentiable than differentiable functions, and since the thermal equilibrium distribution is very spread-out, one might expect that Ψ has probability zero to be differentiable. However, we show that for relevant Hamiltonians the contrary is the case: with probability one, Ψ is infinitely often differentiable and even analytic. We also show that with probability one, Ψ lies in the domain of the Hamiltonian.
We consider the thermal equilibrium distribution at inverse temperature β, or canonical ensemble, of the wave function Ψ of a quantum system. Since L 2 spaces contain more nondifferentiable than differentiable functions, and since the thermal equilibrium distribution is very spread-out, one might expect that Ψ has probability zero to be differentiable. However, we show that for relevant Hamiltonians the contrary is the case: with probability one, Ψ is infinitely often differentiable and even analytic. We also show that with probability one, Ψ lies in the domain of the Hamiltonian.
We consider the thermal equilibrium distribution at inverse temperature β, or canonical ensemble, of the wave function Ψ of a quantum system. Since L2 spaces contain more nondifferentiable than differentiable functions, and since the thermal equilibrium distribution is very spread out, one might expect that Ψ has probability zero to be differentiable. However, we show that for relevant Hamiltonians the contrary is the case: with probability 1, Ψ is infinitely often differentiable and even analytic. We also show that with probability 1, Ψ lies in the domain of the Hamiltonian.
The recent remarkable developments in quantum optics, mesoscopic and cold atom physics have given reality to wave functions. It is then interesting to explore the consequences of assuming ensembles over the wave functions simply related to the canonical density matrix. In this note we analyze a previously introduced distribution over wave functions which naturally arises considering the Schrödinger equation as an infinite dimensional dynamical system. In particular, we discuss the low temperature fluctuations of the quantum expectations of coordinates and momenta for a particle in a double well potential. Our results may be of interest in the study of chiral molecules.
Statistical mechanics reveals that the properties of a macroscopic physical system emerge as an average over an ensemble of statistically independent microscopic subsystems, each occupying a specific microstate. In the study of quantum systems, these microstates can be chosen to correspond to the pure state wave functions of individual quantum systems. However, the physical principles that govern the distribution of a pure state wave function ensemble, even under conditions of thermal equilibrium, are not well established. For instance, the canonical Boltzmann distribution cannot be applied to wave functions because they lack a definite energy. In this paper we present a maximum entropy principle for the quantum wave function ensemble at thermal equilibrium, the so-called Scrooge ensemble. We highlight that a constraint on the energy expectation value, or even the shape of the associated eigenstate distribution, fails to yield a valid equilibrium state. We find that in addition to these constraints, one must also constrain the measurement entropy to be equal to the Rényi divergence of the ensemble with respect to the Gibbs state, indicating that the Rényi divergence may have uninvestigated physical importance to thermal equilibrium in quantum systems.
An open quantum system (i.e., one that interacts with its environment) is almost always entangled with its environment; it is therefore usually not attributed a wave function but only a reduced density matrix ρ. Nevertheless, there is a precise way of attributing to it a wave function ψ1, called its conditional wave function, which is a random wave function of the system whose probability distribution µ1 depends on the entangled wave function ψ ∈ H1 ⊗ H2 in the Hilbert space of system and environment together. We prove several universality (or typicality) results about µ1; they show that if the environment is sufficiently large then µ1 does not depend much on the details of ψ and is approximately given by one of the so-called GAP measures. Specifically, for most entangled states ψ with given reduced density matrix ρ1, µ1 is close to GAP (ρ1). We also show that, if the coupling between the system and the environment is weak, then for most entangled states ψ from a microcanonical subspace corresponding to energies in a narrow interval [E, E + δE] (and most bases of H2), µ1 is close to GAP (ρβ) with ρβ the canonical density matrix on H1 at inverse temperature β = β(E). This provides the mathematical justification of the claim that GAP (ρβ) is the thermal equilibrium distribution of ψ1.
We consider the notion of thermal equilibrium for an individual closed macroscopic quantum system in a pure state, i.e., described by a wave function. The macroscopic properties in thermal equilibrium of such a system, determined by its wave function, must be the same as those obtained from thermodynamics, e.g., spatial uniformity of temperature and chemical potential. When this is true we say that the system is in macroscopic thermal equilibrium (MATE). Such a system may, however, not be in microscopic thermal equilibrium (MITE). The latter requires that the reduced density matrices of small subsystems be close to those obtained from the microcanonical, equivalently the canonical, ensemble for the whole system. The distinction between MITE and MATE is particularly relevant for systems with many-body localization for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. We note, however, that for generic macroscopic systems, including those with MBL, most wave functions in an energy shell are in both MATE and MITE. For a classical macroscopic system, MATE holds for most phase points on the energy surface, but MITE fails to hold for any phase point.
In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered 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+δE. The thermal equilibrium macro‐state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is “close” to Heq. We argue that for “typical” Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann’s quantum ergodic theorem of 1929.