涉及双变数Hermite多项式的二项式定理及其在量子光学中的应用

范洪义; 楼森岳; 潘孝胤; 笪诚

doi:10.7498/aps.63.110304

摘要

我们用量子力学算符Hermite多项式方法发现了涉及双变数Hermite多项式的二项式的两个定理，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. 若干常用的特殊函数的宗量也由普通数变为算符.

关键词:

We propose an operator Hermite polynomial method, namely, to replace the special functions' argument by quantum mechanical operator, and in this way we have derived two binomial theorems related to two-variable Hermite polynomials. This method is concise and may be of help in deducing many operator identities, which may become a new branch in mathematical physics theory.

Keywords:

作者及机构信息

1.
宁波大学物理系, 宁波 315211;

2.
中国科学技术大学材料科学与工程系, 合肥 230026

基金项目: 国家自然科学基金（批准号：11175113）资助的课题.

Authors and contacts

1.
Department of Physics, Ningbo University, Ningbo 315211, China;

2.
Department of Materials Science and Engineering, University of Science and Technology of China, Hefei 230026, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11175113).

参考文献

[1]	Fan H Y, Lou S Y, Pan X Y, Da C 2013 Acta Phys. Sin. 62 240301 (in Chinese) [范洪义, 楼森岳, 潘孝胤, 笪诚 2013 物理学报 62 240301]
[2]	Fan H Y, Zhou J 2012 Science China, Physics, Mechanics m& Astronomy 55 605
[3]	A. Erdelyi 1953 Higher Transcendental Functions (The Bateman Manuscript Project) (McGraw Hill)
[4]	Fan H Y, Jiang T F 2007 Mod. phys. Lett. B 21 475
[5]	Fan H Y, Tang X B 2008 Devolopment of basis of the mathematical physics of quantum mechanics (Hefei:USTC 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)]
[6]	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 物理学报 61 110302]
[7]	Fan H Y, Lu H L, Fan Y 2006 Ann. Phys. 321 480
[8]	Einstein A, Podolsky B, Rosen N 1935 Phys. Rev. 47 777
[9]	Braunstein S L, Loock P V 2005 Rev. Mod. Phys. 77 513
[10]	Fan H Y, Klauder J R 1994 Phys. Rev. A 49 704
[11]	Fan H Y, Zaidi H R, Klauder J R 1987 Phys. Rev. D 35 1831
[12]	Fan H Y 2003 J. Opt. B: Quantum Semiclass. Opt. 5 R147
[13]	Liang X T, Fan H Y 2001 Chin. Phys. B 10 486 (in Chinese) [梁先庭, 范洪义 2001 中国物理 10 486]
[14]	Fan H Y, Da C 2013 Chin. Phys. B 22 090303
[15]	Arrechi F T, Courtons E, Gilmore R, Thomas H 1972 Phys. Rev. A 6 2211
[16]	Schwinger J 1965 Quantum Theory of Angular momentum (New York:Academic press)
[17]	Fan H Y, Chen J H 2003 Eur. Phys. J. D 23 437

施引文献

[1]	Fan H Y, Lou S Y, Pan X Y, Da C 2013 Acta Phys. Sin. 62 240301 (in Chinese) [范洪义, 楼森岳, 潘孝胤, 笪诚 2013 物理学报 62 240301]
[2]	Fan H Y, Zhou J 2012 Science China, Physics, Mechanics m& Astronomy 55 605
[3]	A. Erdelyi 1953 Higher Transcendental Functions (The Bateman Manuscript Project) (McGraw Hill)
[4]	Fan H Y, Jiang T F 2007 Mod. phys. Lett. B 21 475
[5]	Fan H Y, Tang X B 2008 Devolopment of basis of the mathematical physics of quantum mechanics (Hefei:USTC 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)]
[6]	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 物理学报 61 110302]
[7]	Fan H Y, Lu H L, Fan Y 2006 Ann. Phys. 321 480
[8]	Einstein A, Podolsky B, Rosen N 1935 Phys. Rev. 47 777
[9]	Braunstein S L, Loock P V 2005 Rev. Mod. Phys. 77 513
[10]	Fan H Y, Klauder J R 1994 Phys. Rev. A 49 704
[11]	Fan H Y, Zaidi H R, Klauder J R 1987 Phys. Rev. D 35 1831
[12]	Fan H Y 2003 J. Opt. B: Quantum Semiclass. Opt. 5 R147
[13]	Liang X T, Fan H Y 2001 Chin. Phys. B 10 486 (in Chinese) [梁先庭, 范洪义 2001 中国物理 10 486]
[14]	Fan H Y, Da C 2013 Chin. Phys. B 22 090303
[15]	Arrechi F T, Courtons E, Gilmore R, Thomas H 1972 Phys. Rev. A 6 2211
[16]	Schwinger J 1965 Quantum Theory of Angular momentum (New York:Academic press)
[17]	Fan H Y, Chen J H 2003 Eur. Phys. J. D 23 437

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