Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

• URL: http://arxiv.org/abs/2210.10018v2
• Date: Mon, 16 Jan 2023 15:56:27 GMT
• Title: Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace
• Authors: Stefan Teufel, Roderich Tumulka, Cornelia Vogel
• Abstract summary: We consider a macroscopic quantum system with unitarily evolving pure state $psi_tin mathcalH$. We prove two facts about the evolution of the superposition weights $|P_nupsi_t|2$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider a macroscopic quantum system with unitarily evolving pure state $\psi_t\in \mathcal{H}$ 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 $\|P_\nu\psi_t\|^2$: 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 $t\mapsto \|P_\nu \psi_t\|^2$ 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 $t\in[0,\infty)$, $\|P_\nu \psi_t\|^2$ 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.

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