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研究算符函数的有序化排列是一项重要的数理任务. 本文利用特殊函数和正规乘积排序与反正规乘积排序间的互换法则法导出了幂算符
$ {\left(a{a}^\dagger \right)}^{\pm n} $ 和$ {\left({a}^\dagger a\right)}^{\pm n} $ 的正规与反正规乘积排序. 进一步, 利用类比法得到了算符$ {\left(XP\right)}^{\pm n} $ 和$ {\left(PX\right)}^{\pm n} $ 的坐标-动量排序与动量-坐标排序式. 最后, 对新得到的这些算符结果的应用进行一些讨论.Operator ordering is often fallen back on due to its convenience in quantum optics and quantum statistics, thus it is an important task to derive the various ordered forms of operators as directly as possible. In this paper we arrange quantum mechanical operators$ {\left(a{a}^\dagger \right)}^{\pm n} $ and$ {\left({a}^\dagger a\right)}^{\pm n} $ in their normally and anti-normally ordered product forms by using special functions and general mutual transformation rules between normal and anti-normal orderings of operators. Furthermore, the Q- and P-ordered forms of power operators$ {\left(XP\right)}^{\pm n} $ and$ {\left(PX\right)}^{\pm n} $ are also obtained by the analogy method. Finally, some applications are discussed, such as the Glauber-Sudarshan$ P $ -representation of chaotic light field and the generating functions of even and odd bivariate Hermite polynomials.-
Keywords:
- ordered product of operators /
- special function /
- mutual transformation rules /
- analogy method
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[1] Glauber R J 1963 Phys. Rev. 131 2766
Google Scholar
[2] Dirac P A M 1985 Principles of Quantum mechanics (Oxford: Oxford University Press) p167
[3] Scully M O, Zubairy M S 1997 Quantum Optics (Cambridge: Cambridge University Press) p354
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Google Scholar
[5] Fan H Y 2012 Sci. China-Phys. Mech. Astron. 55 762
Google Scholar
[6] Lee H W 1995 Phys. Rep. 259 147
Google Scholar
[7] Balazs N L, Jenning B K 1984 Phys. Rep. 104 347
Google Scholar
[8] Wang J S, Fan H Y, Meng X G 2012 Chin. Phys. B 21 064204
[9] Meng X G, Wang J S, Liang B L 2009 Chin. Phys. B 18 1534
Google Scholar
[10] 徐世民, 张运海, 徐兴磊 2012 大学物理 31 1
Google Scholar
Xu S M, Zhang Y H, Xu X L 2012 Coll. Phys. 31 1
Google Scholar
[11] 范洪义, 楼森岳, 张鹏飞 2015 物理学报 64 160302
Google Scholar
Fan H Y, Lou S Y, Zhang P F 2015 Acta Phys. Sin. 64 160302
Google Scholar
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Google Scholar
[14] Mansour T, Matthias S 2016 Commutation Relations, Normal Ordering, and Stirling Numbers (Boca Raton: CRC Press) pp51−65
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Google Scholar
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Google Scholar
[18] Klauder J R, Sudarshan E C G 1968 Fundamentals of Quantum Optics (New York: W A Benjamin) p279
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