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把量子力学与数理统计的正态分布联系起来进行初步的尝试. 用数理统计的观点和有序算符内的积分技术研究相干态，指出在依赖一个实参数k的量子化方案中，相干态|z >z| 在相空间呈现出以（q，p）为随机变量的两维正态分布，z=（q+ip）/√2. 两个随机变量的相关系数为ik. 在k=±1 的参数相空间中，|z >z|分别表现出 P排序（P在Q左）和Q排序的形式（Q 在P 左），而在k=0的参数相空间中，|z >z|表现出Weyl排序的形式. 在 P 排序和Q排序的情况下，量子算符|z >z||z=（q+ip）/√2的经典对应函数中随机变量（q，p）是关联的，只有在Weyl 对应时，随机变量（q，p）是独立的. 也就是说，算符的Weyl排序有利于其经典对应的随机变量解脱关联.Combining quantum mechanics and the normal distribution in statistics we study the coherent state from the point of view of statistics and by using the integration method within ordered product of operators. We find that the pure coherent state |z >z| exhibits a bivariate normal distribution of randon variables in (q,p) phase space, z=(q+ip)/√2, with a real k-parameter which is related to the quantization scheme, and the correlation coefficient is ik. For k=±1, |z >z| respectively is arranged as P-ordering (all P stand on the left of all Q) and Q-ordering (all Q stand on the left of all P), while in the case of k=0, |z >z| is arranged as the Weyl-ordering. In the cases of P-ordering and Q-ordering, in the classical correspondence function of |z >z||z=(q+ip)/√2 the bivariates (q,p) are correlated, only in the case of Weyl correspondece, (q,p) are independent. In other words, the Weyl ordering of operators is liable to decouple the correlation in bivariates.
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Keywords:
- normal distribution /
- coherent state /
- phase space /
- correlation coefficient
[1] Dirac P A M 1930 The Principle of Quantum Mechanics (Oxford: Clarendon Press)
[2] Fan H Y 2012 Representation and Transformation Theory in Quantum Mechanics-Progress of Dirac’s Symbolic Method (2nd Ed.) (Hefei: USTC Press) (in Chinese) [范洪义 2012 量子力学表象与变换论——狄拉克符号法进展 (合肥: 中国科学技术大学出版社)]
[3] Fan H Y, Lu H L, Fan Y 2006 Ann. Phys. 321 480
[4] Zhang X Y, Wang J S 2011 Acta Phys. Sin. 60 090304 (in Chinese) [张晓燕, 王继锁 2011 60 090304]
[5] Li H Q, Meng X G, Wang J S 2008 Chin. Phys. B 17 2973
[6] Weyl H 1927 Z. Phys. 46 1
[7] Wigner E 1932 Phys. Rev. 40 749
[8] Fan H Y 1992 J. Phys. A 25 3443
[9] Fan H Y 2006 Ann. Phys. 323 1502
[10] Glauber R J 1963 Phy. Rev. 130 2529
[11] Zhou G R 1984 Probability and Statistics (Beijing: Higher Education Press) (in Chinese) [周概容 1984 概率论与数理统计 (北京: 高等教育出版社)]
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[1] Dirac P A M 1930 The Principle of Quantum Mechanics (Oxford: Clarendon Press)
[2] Fan H Y 2012 Representation and Transformation Theory in Quantum Mechanics-Progress of Dirac’s Symbolic Method (2nd Ed.) (Hefei: USTC Press) (in Chinese) [范洪义 2012 量子力学表象与变换论——狄拉克符号法进展 (合肥: 中国科学技术大学出版社)]
[3] Fan H Y, Lu H L, Fan Y 2006 Ann. Phys. 321 480
[4] Zhang X Y, Wang J S 2011 Acta Phys. Sin. 60 090304 (in Chinese) [张晓燕, 王继锁 2011 60 090304]
[5] Li H Q, Meng X G, Wang J S 2008 Chin. Phys. B 17 2973
[6] Weyl H 1927 Z. Phys. 46 1
[7] Wigner E 1932 Phys. Rev. 40 749
[8] Fan H Y 1992 J. Phys. A 25 3443
[9] Fan H Y 2006 Ann. Phys. 323 1502
[10] Glauber R J 1963 Phy. Rev. 130 2529
[11] Zhou G R 1984 Probability and Statistics (Beijing: Higher Education Press) (in Chinese) [周概容 1984 概率论与数理统计 (北京: 高等教育出版社)]
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