ERGODIC THEOREM IN QUANTUM MECHANICS

M. Howard Lee

The physical theory for the ergodic hypothesis is premised on the idea that the hypothesis is measurable by scattering experiment. Therewith it proves the hypothesis by measurable properties of the response function. Birkhoff's theorem proves the ergodic hypothesis by abstract properties of phase space, the measure and transitivity. In this work we apply the two independent approaches to a particular many-body model (spin-1/2 XY model) to see whether they arrive at the same conclusions on the ergodicity question. The two approaches are compared to gain insight into their goals.

Julio Felix Valdez Autonell, Walter Enrique Lebrecht Diaz-Pinto, Jaime Octavio Cartes Muller

Giovanni Maria Prosperi, Alexandria Scotti

CHANG TSUNG-SUI, 中国科学院数学研究所

In a paper by Klein, it is pointed out that for the quantum mechanical ergodic theorem (time average = average over states) to be valid, all integrals of motion R must satisfy the condition ∑ρ"(α′ρ"|R|β′ρ")=const δα′β′ (1) where α′, β′, γ, … refer to the assembly under question and α", β",…, ρ" ,… refer to the surrounding with which our assembly is in equilibrium. In this short note, it is pointed out that the argument of Klein is doubtful at one point and a slightly different approach is given. In this approach, it is shown that (1) is actually sufficient, but only after introducing an additional assumption that the different states of the surroundings have equal probabilities, and that in general, a stronger condition such as (α′ρ″|R|β′θ″)=const δα′β′δρ″θ″ (2) is needed. Cases with integrals of motion is also considered. We consider first an isolated system, write the wave function as (α1α2>, where α1 represents a set of commuting integrals of motion and α2 a set of observables not containing integrals of motion and introduce the assumption that for any integral of motion R, the matrix 1α′2Rα1"α2"> representing it is of the form φ(α′1α1")δ(α′2α2"). (3) Under such conditions, we prove that if the initial state of our system is an eigenstate of α1 corresponding to the eigenvalue α10, then the average of an observable F over time is given by ∑α210α2|F|α10α2>/∑α210α2|1|α10α2>, a result which is clearly to be expected. Extension to systems interacting with external surroundings is easily made.