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提出量子力学算符Hermite多项式方法，即将若干常用的特殊函数的宗量由普通数变为算符，并用它来发现涉及Hermite多项式（单变数和双变数）的二项式定理和涉及Laguerre多项式的负二项式定理，它们在计算若干量子光场的物理性质时有实质性的应用. 该方法不但具有简捷的优点，而且能导出很多新的算符恒等式，成为发展数学物理理论的一个重要分支.
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关键词:
- 量子力学 /
- Hermite多项式 /
- 二项式定理 /
- Laguerre多项式
We propose an operator Hermite polynomial method, namely, we replace the arguments of the special function by quantum mechanical operators, and in this way we derive a binomial theorem involving Hermite polynomials and a negative-binomial theorem involving Laguerre polynomials. These two theorems will have essential applications in quantum optics calculations. This method is concise and helpful in deducing many operator identities, which may become a new branch in mathematical physics theory.-
Keywords:
- quantum mechanics /
- Hermite polynoms /
- binomial theorem /
- Laguerre polynomials
[1] Fan H Y, Ruan T N 1985 Commun. Theor. Phys. 4 181
[2] Fan H Y, Ruan T N 1989 Commun. Theor. Phys. 12 219
[3] Fan H Y 2012 Representation and Transformation Theory in Quantum Mechanics-Progress of Dirac’s Symbolic Method (2nd Ed.) (Hefei: University of Science and Technology of China 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)]
[4] Fan H Y, Hu L Y 2010 Optical Transformation-From Quantum to Classical (Shanghai: Shanghai Jiao Tong University Press) (in Chinese) [范洪义, 胡利云 2010 光学变换——从量子到经典 (上海: 上海交通大学出版社)]
[5] Fan H Y, Li X C 2012 Acta Phys. Sin. 61 200301 (in Chinese) [范洪义, 李学超 2012 61 200301]
[6] Erdelyi A 1953 Higher Transcendental Functions (The Bateman Manuscript Project) (New York: McGraw Hill)
[7] Fan H Y, Jiang T F 2007 Mod. Phys. Lett. B 21 475
[8] Fan H Y, Tang X B 2008 Devolopment of Basis of the Mathematical Physics of Quantum Mechanics (Hefei: University of Science and Technology of China Press) (in Chinese) [范洪义, 唐绪兵 2008 量子力学数理基础进展 (合肥: 中国科学技术大学出版社)]
[9] Fan H Y, Zhan D H, Yu W J, Zhou J 2012 Acta Phys. Sin. 61 110302 (in Chinese) [范洪义, 展德会, 于文健, 周军 2012 61 110302]
[10] Fan H Y 2003 J. Opt. B: Quantum Semiclass. Opt. 5 R147
[11] Fan H Y, Lu H L, Fan Y 2006 Ann. Phys. 321 480
[12] Fan H Y, L C H 2012 The Phase Space Theory of Quantum Mechanics (Shanghai: Shanghai Jiao Tong University Press) (in Chinese) [范洪义, 吕翠红 2012 量子力学的相空间理论 (上海: 上海交通大学出版社)]
[13] Fan H Y, Zaidi H R, Klauder J R 1987 Phys. Rev. D 35 1831
[14] Fan H Y, Da C 2013 Chin. Phys. B 22 090303
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[1] Fan H Y, Ruan T N 1985 Commun. Theor. Phys. 4 181
[2] Fan H Y, Ruan T N 1989 Commun. Theor. Phys. 12 219
[3] Fan H Y 2012 Representation and Transformation Theory in Quantum Mechanics-Progress of Dirac’s Symbolic Method (2nd Ed.) (Hefei: University of Science and Technology of China 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)]
[4] Fan H Y, Hu L Y 2010 Optical Transformation-From Quantum to Classical (Shanghai: Shanghai Jiao Tong University Press) (in Chinese) [范洪义, 胡利云 2010 光学变换——从量子到经典 (上海: 上海交通大学出版社)]
[5] Fan H Y, Li X C 2012 Acta Phys. Sin. 61 200301 (in Chinese) [范洪义, 李学超 2012 61 200301]
[6] Erdelyi A 1953 Higher Transcendental Functions (The Bateman Manuscript Project) (New York: McGraw Hill)
[7] Fan H Y, Jiang T F 2007 Mod. Phys. Lett. B 21 475
[8] Fan H Y, Tang X B 2008 Devolopment of Basis of the Mathematical Physics of Quantum Mechanics (Hefei: University of Science and Technology of China Press) (in Chinese) [范洪义, 唐绪兵 2008 量子力学数理基础进展 (合肥: 中国科学技术大学出版社)]
[9] Fan H Y, Zhan D H, Yu W J, Zhou J 2012 Acta Phys. Sin. 61 110302 (in Chinese) [范洪义, 展德会, 于文健, 周军 2012 61 110302]
[10] Fan H Y 2003 J. Opt. B: Quantum Semiclass. Opt. 5 R147
[11] Fan H Y, Lu H L, Fan Y 2006 Ann. Phys. 321 480
[12] Fan H Y, L C H 2012 The Phase Space Theory of Quantum Mechanics (Shanghai: Shanghai Jiao Tong University Press) (in Chinese) [范洪义, 吕翠红 2012 量子力学的相空间理论 (上海: 上海交通大学出版社)]
[13] Fan H Y, Zaidi H R, Klauder J R 1987 Phys. Rev. D 35 1831
[14] Fan H Y, Da C 2013 Chin. Phys. B 22 090303
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