Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

Abstract

We consider a macroscopic quantum system with unitarily evolving pure state psi _tin mathcal {H} and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces mathcal {H}_nu (macro spaces) of mathcal {H}. Let P_nu denote the projection to mathcal {H}_nu . We prove two facts about the evolution of the superposition weights Vert P_nu psi _tVert ^2: First, given any T>0, for most initial states psi _0 from any particular macro space mathcal {H}_mu (possibly far from thermal equilibrium), the curve tmapsto Vert P_nu psi _tVert ^2 is approximately the same (i.e., nearly independent of psi _0) on the time interval [0, T]. And second, for most psi _0 from mathcal {H}_mu and most tin [0,infty ), Vert P_nu psi _tVert ^2 is close to a value M_{mu nu } that is independent of both t and psi _0. The first is an instance of the phenomenon of dynamical typicality observed by Bartsch, Gemmer, and Reimann, and the second modifies, extends, and in a way simplifies the concept, introduced by von Neumann, now known as normal typicality.