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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.
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Keywords:
- composite function operator /
- differential quotient rule /
- Wigner operator /
- ordered product of operators
[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 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 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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[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 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 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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