量子混沌系统本征态的统计非遍历性及其半经典极限

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

讨论半经典极限下ｃｉｒｃｕｌａｒ ｕｎｉｔａｒｙ ｅｎｓｅｍｂｌｅ（ＣＵＥ） 系综本征态θｋ（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ρｋ（ｊ） 的一对统计函数：ΦＮ（ｊ）＝ ∑Ｎ－１ｋ＝ ０ρｋ（ｊ）２ 和ΨＮ（ｊ） ＝ ∑Ｎ－１ｋ＝ ０ρｋ（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随机矢量得到的ΦＮ（ｊ） 和 ΨＮ（ｊ） ，发现它们的平均值和涨落随Ｎ 的增大而趋于０ ，并且有明显的标度关系．与由量子面包师变换的本征态得到的ΦＮ（ｊ） 和Ψｎ（ｊ） 的结果相比较，发现疤痕的存在使得量子面包师变换的本征态的统计函数的涨落比随机矩阵的统计函数的涨落要大，并且在Ｎ→∞的半经典极限下，以较慢的速度趋于０ ．

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→∞ . 