约束引起的算符次序问题及其解决
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摘要
微分几何中的曲面论,用两个变量就可以完全参数化一个二维曲面。也就是,当一个粒子约束在曲面上运动时,只需要两个独立的变量就可以完全刻划粒子的运动。但是,当我们从物理的角度审视这一问题时,就会发现可以把这个曲面嵌入一个三维空间中,粒子的运动可以在笛卡儿坐标下分解为三个互相正交的方向。动能算符T和三个笛卡儿动量Pi (i =1,2,3)的关系似乎为: 其中Pi (i =1,2,3)为厄密算符。事实上,这一关系的成立是有条件的:要么不存在约束,要么发生在经典极限下。在存在约束时,上式应代之以其中fi (i =1,2,3)是非平凡的函数。本文用圆环面,旋转抛物面,旋转单叶双曲面,磁场中的荷电平面转子,球面转子等体系中的量子运动说明函数fi(i =1,2,3)是存在的。
在不同的矢势下,荷电粒子的力学动量从而动能的表达式是不同的。本文研究了它们之间和量子规范相因子的关系,发现规范相因子会自然出现在动能算符中。
According to surface theory in differential geometry, the two-dimensional surface is parameterized by two variables. I. e., when a particle moves on the surface, only two variables suffice to describe the motion of the particle. However, when examining the same problem in the physical point of view, the motion of the particle can also be described in the three-dimensional coordinates. Explicitly, the relation between kinetic energy T and Hermitian Cartesian momentum Pi ( i=1,2,3)is speciously which holds either for the system being free of constraint or for the system in classical limit. Moreover, under constraint, above expression should be replaced by, where fi ( i=1,2,3)are the non-trivial functions. This paper utilizes quantum motions on the torus surface, paraboloid of revolution, hyperboloid of revolution of one sheet, charged planar and spherical rotator etc., to demonstrate the existence of the function f i.
Since different vector potentials lead to different kinetic energies, the relationship between these kinetic energies and the gauge phase factors, is studied. Results show that the gauge phase factors can appear in the kinetic operator naturally.
在不同的矢势下,荷电粒子的力学动量从而动能的表达式是不同的。本文研究了它们之间和量子规范相因子的关系,发现规范相因子会自然出现在动能算符中。
According to surface theory in differential geometry, the two-dimensional surface is parameterized by two variables. I. e., when a particle moves on the surface, only two variables suffice to describe the motion of the particle. However, when examining the same problem in the physical point of view, the motion of the particle can also be described in the three-dimensional coordinates. Explicitly, the relation between kinetic energy T and Hermitian Cartesian momentum Pi ( i=1,2,3)is speciously which holds either for the system being free of constraint or for the system in classical limit. Moreover, under constraint, above expression should be replaced by, where fi ( i=1,2,3)are the non-trivial functions. This paper utilizes quantum motions on the torus surface, paraboloid of revolution, hyperboloid of revolution of one sheet, charged planar and spherical rotator etc., to demonstrate the existence of the function f i.
Since different vector potentials lead to different kinetic energies, the relationship between these kinetic energies and the gauge phase factors, is studied. Results show that the gauge phase factors can appear in the kinetic operator naturally.
引文
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[4] 李政道. 场论与粒子物理学(上册). 北京: 科学出版社,1980, 7(1)
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[8] Dirac P A M. Generalize Hamiltonian Dynamics. Can J Math, 1950, 2:129-143
[9] DeWitt B S. Dynamical Theory in Curved Spaces. Rev Mod Phys, 1967, 29: 377
[10]Cheng K S. Improved Dirac quantization of a free particle. J Math Phys, 1972, 13: 1723
[11]Kleinert H. Quantum Equivalence Principle for Path Integrals in Spaces with Curvature and Torsion. Mod Phys Lett A, 1989, 4: 2329
[12]Kleinert H. Classical and Fluctuating Paths in Spaces with Curvature and Torsion. Phys. Lett B, 1990, 236: 315
[13]Kleinert Hagen. Path Integrals in Quantum Mechanics Statistics and Polymer Physics. Vol 2. Singapore: World Scientific, 2003
[14]Borovitskaya E , Shur M S. Quantum Dots. Singapore: World scientific Publishing, 2002
[15]Turton R. The Quantum Dot: A Journey into the Future of Microelectronics. London: Oxford Univ, 1996
[16]Londergan J T, Carini J P, Murdock D P. Binding and Scattering in Two Dimensional Systems: Applications to Quantum Wires, Waveguides, and Photonic Crystals. Berlin: Springer-Verlag, 1999
[17]Ferry D, Goodnick S. Transport in Nanostructures. Cambridge: Cambridge University Press, 1992
[18]Midgley S, Wang J B. Time-dependent quantum waveguide theory: a study of nano ring structures. Aust J Phys, 2000, 53: 77-85
[19]Bouwmeester D, Ekert A, Zeilinger A. The Physics of Quantum Information. Berlin: Springer-Verlag, 2000
[20]张永德,吴盛俊,侯广等. 量子信息论物理原理和某些进展. 2002 年 11 月第一版, 武汉: 华中师范大学出版社, 2002,11
[21] Hirvensalo M. Quantum Computing. New York: Springer-verlag, 2001
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[23]Bennett C H, Brassard G. “Event-ready-detectors” Bell experiment via entanglement swapping. Phys Rev Lett, 1992, 69: 2881
[24]Leuenberger M N, Flatté M E, Awschalom D D. Teleportation of Electronic Many-Qubit States Encoded in the Electron Spin of Quantum Dots via Single Photons. Phys Rev Lett, 2005, 94: 107401
[25] Wang L J, Kuzmich A, Dogariu A. Gain -Asisted Superluminal light Propagation. Nature, 2000, 406(13): 277
[26] Huang C G, Zhang Y Z. Poynting vector, energy density, and energy velocity in an anomalous dispersion medium. Phys Rev A, 2001, 65(14): 015802
[27] Huang C G, Zhang Y Z. Propagation of a Rectangular Pulse in an Anomalous Dispersive Medium. J Opt A Pure Appl, 2002, 4: 263-270
[28]Scully M O. Enhancement of the index of refraction via quantum coherence. Phys Rev Lett,1991, 67: 1855
[29]Field J E, Hahn K H., Harris S E. Observation of electromagnetically induced transparency in collisionally broadened lead vapor. Phys Rev Lett, 1991, 67: 3062
[30]Harris S E, Field J E, Kasapi A. Dispersive properties of electromagnetically induced transparency. Phys Rev A, 1992, 46: R29
[31]Cohen T C, Diu B, Lalo? F. Quantum Mechanics. Vol 2, New York: Hermann and John Wiley and Sons, 1977
[32]Liu Q H, Liu T G. Quantum Hamiltonian for the Rigid Rotator. International Journal of Theoretical Physics, 2003, 42(12):2877-2880
[33]刘全慧,刘天贵.在三维直角坐标系中研究转子体系的动能表达式.大学物理,2004, 23(3): 32-33
[34]Liu Q H, Hou J X, Xiao Y P, et al. Quantum Motion on 2D surface of Nonspherical Topology. International of Theoretical Physics, 2004, 43:1011
[35]Encinosa M, Mott L. Curvature-induced toroidal bound states. Phys Rev A, 2003, 68:014102
[36]王竹溪,郭敦仁. 特殊函数概论. 北京:科学出版社,1979
[37]Jackson J D. Classical Electrodynamics. New York: John Wiley and Sons, 1975
[38]杨振宁.磁单极理论的进展, 见《第 19 次国际高能物理会议录》. 1978: 497-500
[39]贾艳伟,刘全慧,彭解华.氢原子的 1 r矩阵元的量子-经典对应.物理学报,2002,51(2):0365-0368
[40] 喀兴林.谈谈量子力学中的算符.量子力学朝花夕拾—教与学篇 (王文正,柯善哲,刘全慧主编).2004,10(1):46-54
[2] Podolsky B. Quantum-Mechanically Correct Form of Hamiltonian Function For Conservative Systems. Physics Review, 1928, 32(5): 812-816
[3] 席夫 L I. 量子力学. 李淑娴,陈崇光译. 北京: 人民教育出版社,1981,9(1), 200
[4] 李政道. 场论与粒子物理学(上册). 北京: 科学出版社,1980, 7(1)
[5] Woodhouse. N M. Geometric Quantization. Oxford: Oxford University Press, 1992
[6] Dirac P A M. Magnetic monopole. in Directions of Physics eds Hora, J Shepanski, Jone Wiley and Sons, New York ,1978
[7] Dirac P A M. Lectures on Quantum Mechanics. New York: Yeshiva university, 1964
[8] Dirac P A M. Generalize Hamiltonian Dynamics. Can J Math, 1950, 2:129-143
[9] DeWitt B S. Dynamical Theory in Curved Spaces. Rev Mod Phys, 1967, 29: 377
[10]Cheng K S. Improved Dirac quantization of a free particle. J Math Phys, 1972, 13: 1723
[11]Kleinert H. Quantum Equivalence Principle for Path Integrals in Spaces with Curvature and Torsion. Mod Phys Lett A, 1989, 4: 2329
[12]Kleinert H. Classical and Fluctuating Paths in Spaces with Curvature and Torsion. Phys. Lett B, 1990, 236: 315
[13]Kleinert Hagen. Path Integrals in Quantum Mechanics Statistics and Polymer Physics. Vol 2. Singapore: World Scientific, 2003
[14]Borovitskaya E , Shur M S. Quantum Dots. Singapore: World scientific Publishing, 2002
[15]Turton R. The Quantum Dot: A Journey into the Future of Microelectronics. London: Oxford Univ, 1996
[16]Londergan J T, Carini J P, Murdock D P. Binding and Scattering in Two Dimensional Systems: Applications to Quantum Wires, Waveguides, and Photonic Crystals. Berlin: Springer-Verlag, 1999
[17]Ferry D, Goodnick S. Transport in Nanostructures. Cambridge: Cambridge University Press, 1992
[18]Midgley S, Wang J B. Time-dependent quantum waveguide theory: a study of nano ring structures. Aust J Phys, 2000, 53: 77-85
[19]Bouwmeester D, Ekert A, Zeilinger A. The Physics of Quantum Information. Berlin: Springer-Verlag, 2000
[20]张永德,吴盛俊,侯广等. 量子信息论物理原理和某些进展. 2002 年 11 月第一版, 武汉: 华中师范大学出版社, 2002,11
[21] Hirvensalo M. Quantum Computing. New York: Springer-verlag, 2001
[22]李承祖, 黄明球, 陈平形等. 量子通信和量子计算. 2000 年 8 月第一版, 长沙: 国防科技大学出版社, 2001,7
[23]Bennett C H, Brassard G. “Event-ready-detectors” Bell experiment via entanglement swapping. Phys Rev Lett, 1992, 69: 2881
[24]Leuenberger M N, Flatté M E, Awschalom D D. Teleportation of Electronic Many-Qubit States Encoded in the Electron Spin of Quantum Dots via Single Photons. Phys Rev Lett, 2005, 94: 107401
[25] Wang L J, Kuzmich A, Dogariu A. Gain -Asisted Superluminal light Propagation. Nature, 2000, 406(13): 277
[26] Huang C G, Zhang Y Z. Poynting vector, energy density, and energy velocity in an anomalous dispersion medium. Phys Rev A, 2001, 65(14): 015802
[27] Huang C G, Zhang Y Z. Propagation of a Rectangular Pulse in an Anomalous Dispersive Medium. J Opt A Pure Appl, 2002, 4: 263-270
[28]Scully M O. Enhancement of the index of refraction via quantum coherence. Phys Rev Lett,1991, 67: 1855
[29]Field J E, Hahn K H., Harris S E. Observation of electromagnetically induced transparency in collisionally broadened lead vapor. Phys Rev Lett, 1991, 67: 3062
[30]Harris S E, Field J E, Kasapi A. Dispersive properties of electromagnetically induced transparency. Phys Rev A, 1992, 46: R29
[31]Cohen T C, Diu B, Lalo? F. Quantum Mechanics. Vol 2, New York: Hermann and John Wiley and Sons, 1977
[32]Liu Q H, Liu T G. Quantum Hamiltonian for the Rigid Rotator. International Journal of Theoretical Physics, 2003, 42(12):2877-2880
[33]刘全慧,刘天贵.在三维直角坐标系中研究转子体系的动能表达式.大学物理,2004, 23(3): 32-33
[34]Liu Q H, Hou J X, Xiao Y P, et al. Quantum Motion on 2D surface of Nonspherical Topology. International of Theoretical Physics, 2004, 43:1011
[35]Encinosa M, Mott L. Curvature-induced toroidal bound states. Phys Rev A, 2003, 68:014102
[36]王竹溪,郭敦仁. 特殊函数概论. 北京:科学出版社,1979
[37]Jackson J D. Classical Electrodynamics. New York: John Wiley and Sons, 1975
[38]杨振宁.磁单极理论的进展, 见《第 19 次国际高能物理会议录》. 1978: 497-500
[39]贾艳伟,刘全慧,彭解华.氢原子的 1 r矩阵元的量子-经典对应.物理学报,2002,51(2):0365-0368
[40] 喀兴林.谈谈量子力学中的算符.量子力学朝花夕拾—教与学篇 (王文正,柯善哲,刘全慧主编).2004,10(1):46-54