量子混沌系统本征态的统计非遍历性及其半经典极限

讨论半经典极限下circular unitary ensemble(CUE)系综本征态θ_k(j)的非遍历性质.为研究量子系统本征态在统计上的非遍历性，定义了本征态分布ρ_k(j)的一对统计函数：Φ_N(j)=∑N-1k=0ρ_k(j)²和Ψ_N(j)=∑N-1k=0ρ_k(j)，以分别体现量子本征态分布的凸起与凹陷.在随机矩阵理论的框架内，数值地计算了由正交归一的 关键词：

THE STATISTICAL NON-ERGODICITY OF THE EIGENSTATES OF THE QUANTUM CHAOTIC SYSTEMS AND ITS SEMI-CLASSICAL LIMIT

In the semi-classical limit, the non-ergodicity of the eigenstates, θ_k(j), of circular unitary ensemble (CUE) are investigated. To study statistically the non-ergodicity of the eigenstates ofaquantumsystem, a pair of statistical functions,Φ_N(j)=∑N-1k=0|θ_k(j)|⁴ and Ψ_N(j)=∑N-1k=0|θ_k(j)|², are defined to show the scars and anti-scars respectively. In the frame of Random Matrix Theory,Φ_N(j)s and Ψ_N(j)s for random orthohormal unit vectors are calculated. It is shown that their averages and fluctuations will tend to zero with the increase of N,while they follow the scaling laws. Compared with Φ_N(j)s and Ψ_N(j)s obtained from the eigenstates of the quantum baker's transformation, it is found that, with the presence of scars (or antiscars), the fluctuations of the statistical functions of the eigenstates of the quantum baker's transformation will be greater than those of the random matrices, and tend to zero much slower in the semi-classical limit of N→∞.