In the semi-classical limit, the non-ergodicity of the eigenstates, θk(j), of circular unitary ensemble (CUE) are investigated. To study statistically the non-ergodicity of the eigenstates ofaquantumsystem, a pair of statistical functions,ΦN(j)=∑N-1k=0|θk(j)|4 and ΨN(j)=∑N-1k=0|θk(j)|2, are defined to show the scars and anti-scars respectively. In the frame of Random Matrix Theory,ΦN(j)s and ΨN(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 ΦN(j)s and ΨN(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→∞.
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