论文标题
宏观希尔伯特子空间的典型纯状态的时间演变
Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace
论文作者
论文摘要
我们考虑一个具有单位发展的纯状状态$ψ_t\ in \ Mathcal {H} $的宏观量子系统，并认为不同的宏观状态对应于相互正交的，高维的子空间$ \ MATHCAL {H}_ν$（MACRO SPACE）的$ \ MATHCAL} $。令$p_ν$表示投影对$ \ MATHCAL {H}_ν$。我们证明了两个事实有关叠加权重的演变$ \ | | | | | | | | | |^2 $：首先，给定任何$ t> 0 $，对于大多数初始状态，从任何特定的宏空间$ \ mathcal $ \ mathcal {h}_μ$中$ψ_0美元（即，几乎独立于$ψ_0$）在时间间隔$ [0，t] $上。其次，对于大多数$ \ \ \ m varycal {h} _ $和[0，\ infty）$，$ \ |p_νψ_t\ |^2 $的$ \ m narcal {h} _ $而言，大多数$ t \ 2 $都接近$ $ m_ {μν} $，与$ t $ t $ t $ t $和$ψ_0$无关。第一个是Bartsch，Gemmer和Reimann观察到的动态典型性现象的实例，第二次修改，扩展和以某种方式简化了概念，该概念是由Von Neumann引入的，现在被称为正常典型性。
We consider a macroscopic quantum system with unitarily evolving pure state $ψ_t\in \mathcal{H}$ and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces $\mathcal{H}_ν$ (macro spaces) of $\mathcal{H}$. Let $P_ν$ denote the projection to $\mathcal{H}_ν$. We prove two facts about the evolution of the superposition weights $\|P_νψ_t\|^2$: First, given any $T>0$, for most initial states $ψ_0$ from any particular macro space $\mathcal{H}_μ$ (possibly far from thermal equilibrium), the curve $t\mapsto \|P_νψ_t\|^2$ is approximately the same (i.e., nearly independent of $ψ_0$) on the time interval $[0,T]$. And second, for most $ψ_0$ from $\mathcal{H}_μ$ and most $t\in[0,\infty)$, $\|P_νψ_t\|^2$ is close to a value $M_{μν}$ that is independent of both $t$ and $ψ_0$. 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.
