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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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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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[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
[4] Xu S M, Zhang Y H, Xu X L, Li H Q, Wang J S 2016 Chin. Phys. B 25 120301
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 64 160302
Google Scholar
Fan H Y, Lou S Y, Zhang P F 2015 Acta Phys. Sin. 64 160302
Google Scholar
[12] Fan H Y, Wang Z 2004 Chin. Phys. B 23 080301
[13] Xu S M, Zhang Y H, Xu X L, Li H Q, Wang J S 2020 Int. J. Theor. Phys. 59 539
Google Scholar
[14] Mansour T, Matthias S 2016 Commutation Relations, Normal Ordering, and Stirling Numbers (Boca Raton: CRC Press) pp51−65
[15] Lambert F, Springael J 2008 Acta. Appl. Math. 102 147
Google Scholar
[16] Erdelyi A 1953 Higher Transcendental Function: The Batemann Manuscript Project (New York: McGraw-Hill) pp57−62
[17] Hu L Y, Fan H Y 2009 Chin. Phys. B 18 1061
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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