复合函数算符的微商法则及其在量子物理中的应用

徐世民; 徐兴磊; 李洪奇; 王继锁

doi:10.7498/aps.63.240302

摘要

给出了在量子物理学、量子统计学、算符排序理论、矩阵论以及控制理论中有着重要用途的复合函数算符的一般微分法则, 利用这一法则研究了Wigner算符和Weyl对应规则中的积分问题, 证明了两类典型的算符恒等公式. 给出了Wigner算符的有序算符内的微分形式, 并得到了一些重要函数的新的微分式. 最后, 引入了一个参数型的Wigner算符来统一正规序、Weyl编序以及反正规序三种算符排序.

关键词:

Differential quotient rule of composite function operator and its applications in quantum physics, quantum statistics, operator ordering theory, matrix theory and control theory are given. The integration problem of Wigner operator and Weyl corresponding rules are studied. Two kinds of typical operator identity formulas are proved. The differential form of Wigner operator in ordered product of operators and new differential form of important functions are obtained. Finally, a Wigner operator with parameter for unifying regular order, Weyl sequencing and abnormal order is introduced.

Keywords:

作者及机构信息

1.
菏泽学院物理与电子工程系, 菏泽 274015;

2.
菏泽学院量子信息与物理计算重点实验室, 菏泽 274015;

3.
曲阜师范大学物理工程学院, 曲阜 273165

基金项目: 国家自然科学基金(批准号: 11244005)和山东省自然科学基金(批准号: Y2008A16)资助的课题.

Authors and contacts

1.
Department of Physics and Electronic Engineering, Heze University, Heze 274015, China;

2.
Key Laboratory of Quantum Communication and Calculation, Heze University, Heze 274015, China;

3.
College of Physics and Engineering Qufu Normal University, Qufu 273165, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11244005) and the Natural Science Foundation of Shandong Province, China (Grant No. Y2008A16).

参考文献

[1]	Weyl H 1931 The Theory of Group and Quantum Mechanics (New York: Dover Publisher) p272
[2]	Wigner E 1932 Phys. Rev. 40 749
[3]	Fang B R, Zhou J D, Li Y M 2005 Matrix Theory (Beijing: Tsinghua University Press) p218 (in Chinese) [方保镕, 周继东, 李医民 2005 矩阵论 (北京: 清华大学出版社) 第218 页]
[4]	Zhang K Y, Xu Z 2013 Matrix Theory (Beijing: Science Press) p114 (in Chinese) [张凯院, 徐仲2013 矩阵论 (北京: 科学出版社) 第114页]
[5]	Cheng Y P, Zhang K Y, Xu Z 2006 Matrix Theory (Xian: Northwestern Polytechnical University Press) p164 (in Chinese) [程云鹏, 张凯院, 徐仲 2006 矩阵论 (西安: 西北工业大学出版社) 第164页]
[6]	Tian W H 2012 Modern Control Theory (Beijing: People's Posts and Telecommunications Press) p27 (in Chinese) [田卫华 2012 现代控制理论 (北京: 人民邮电出版社) 第27页]
[7]	Zou B M 2007 Automatic Control Theory (Chengdu: Mechanical Industry Press) (in Chinese) p19 (in Chinese) [邹伯敏 2007自动控制理论 (成都: 机械工业出版社) 第19页]
[8]	Fan H Y, Zaidi H R, Klauder J R 1987 Phys. Rev. D 35 1831
[9]	Fan H Y 2003 J. Opt. B: Quantum Semicl Opt. 5 R147
[10]	Wnsche A 1999 J. Opt. B: Quantum Semicl Opt. 1 R11
[11]	Gasiorowicz S 2006 Quantum Mechanics (3rd Ed.) (Beijing: Higher Education Press) p119 [加西欧洛维茨 2006 量子物理 (第三版, 影印版) (北京: 高等教育出版社) 第119页]
[12]	Kardar M 2010 Statistical Physics of Particles (Cambridge: Cambridge University Press) p293
[13]	Fan H Y 1992 J. Phys. A 25 3443
[14]	Achilles R, Bonfiglioli A 2012 Arch. Hist. Exact. Sci. 66 295
[15]	Fan H Y, Zaidi H R 1987 Phys. Lett. A 124 303
[16]	Hu L Y, Fan H Y 2009 Phys. Rev. A 80 022115
[17]	Eldelyi A 1953 Higher Transcendental Functions (Bateman Manuscript Project) (New York: McGraw-Hill) p1
[18]	Xu S M, Jiang J J, Li H Q, Xu X L 2008 Acta Phys. Sin. 57 7430 (in Chinese) [徐世民, 蒋继健, 李洪奇, 徐兴磊 2008 物理学报 57 7430]
[19]	Xu S M, Xu X L, Li H Q, Wang J S 2009 Acta Phys. Sin. 58 2174 (in Chinese) [徐世民, 徐兴磊, 李洪奇, 王继锁 2009 物理学报 58 2174]
[20]	Xu S M, Xu X L, Li H Q, Wang J S 2009 Chin. Phys. B 18 2129

施引文献

[1]	Weyl H 1931 The Theory of Group and Quantum Mechanics (New York: Dover Publisher) p272
[2]	Wigner E 1932 Phys. Rev. 40 749
[3]	Fang B R, Zhou J D, Li Y M 2005 Matrix Theory (Beijing: Tsinghua University Press) p218 (in Chinese) [方保镕, 周继东, 李医民 2005 矩阵论 (北京: 清华大学出版社) 第218 页]
[4]	Zhang K Y, Xu Z 2013 Matrix Theory (Beijing: Science Press) p114 (in Chinese) [张凯院, 徐仲2013 矩阵论 (北京: 科学出版社) 第114页]
[5]	Cheng Y P, Zhang K Y, Xu Z 2006 Matrix Theory (Xian: Northwestern Polytechnical University Press) p164 (in Chinese) [程云鹏, 张凯院, 徐仲 2006 矩阵论 (西安: 西北工业大学出版社) 第164页]
[6]	Tian W H 2012 Modern Control Theory (Beijing: People's Posts and Telecommunications Press) p27 (in Chinese) [田卫华 2012 现代控制理论 (北京: 人民邮电出版社) 第27页]
[7]	Zou B M 2007 Automatic Control Theory (Chengdu: Mechanical Industry Press) (in Chinese) p19 (in Chinese) [邹伯敏 2007自动控制理论 (成都: 机械工业出版社) 第19页]
[8]	Fan H Y, Zaidi H R, Klauder J R 1987 Phys. Rev. D 35 1831
[9]	Fan H Y 2003 J. Opt. B: Quantum Semicl Opt. 5 R147
[10]	Wnsche A 1999 J. Opt. B: Quantum Semicl Opt. 1 R11
[11]	Gasiorowicz S 2006 Quantum Mechanics (3rd Ed.) (Beijing: Higher Education Press) p119 [加西欧洛维茨 2006 量子物理 (第三版, 影印版) (北京: 高等教育出版社) 第119页]
[12]	Kardar M 2010 Statistical Physics of Particles (Cambridge: Cambridge University Press) p293
[13]	Fan H Y 1992 J. Phys. A 25 3443
[14]	Achilles R, Bonfiglioli A 2012 Arch. Hist. Exact. Sci. 66 295
[15]	Fan H Y, Zaidi H R 1987 Phys. Lett. A 124 303
[16]	Hu L Y, Fan H Y 2009 Phys. Rev. A 80 022115
[17]	Eldelyi A 1953 Higher Transcendental Functions (Bateman Manuscript Project) (New York: McGraw-Hill) p1
[18]	Xu S M, Jiang J J, Li H Q, Xu X L 2008 Acta Phys. Sin. 57 7430 (in Chinese) [徐世民, 蒋继健, 李洪奇, 徐兴磊 2008 物理学报 57 7430]
[19]	Xu S M, Xu X L, Li H Q, Wang J S 2009 Acta Phys. Sin. 58 2174 (in Chinese) [徐世民, 徐兴磊, 李洪奇, 王继锁 2009 物理学报 58 2174]
[20]	Xu S M, Xu X L, Li H Q, Wang J S 2009 Chin. Phys. B 18 2129

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