Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

We consider a macroscopic quantum system with unitarily evolving pure state ψt∈ H and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces Hν (macro spaces) of H. Let Pν denote the projection to Hν. We prove two facts about the evolution of the superposition weights ‖ Pνψt‖ 2: First, given any T> 0 , for most initial states ψ from any particular macro space Hμ (possibly far from thermal equilibrium), the curve t↦ ‖ Pνψt‖ 2 is approximately the same (i.e., nearly independent of ψ) on the time interval [0, T]. And second, for most ψ from Hμ and most t∈ [0 , ∞) , ‖ Pνψt‖ 2 is close to a value Mμν that is independent of both t and ψ. 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.