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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 物理学报 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 物理学报 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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