约束在弯曲曲面上运动的粒子动量和动能算符
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摘要
约束在量子点上的量子运动是目前凝聚态物理研究的一个热点。一个很有意思的问题是量子点的几何形状,除了典型的球形和椭球形外,还有面包圈和M?bius环等形状。这些研究提出了一个基础性的研究课题:在经典力学中,对约束在曲面上粒子运动的描述可以在内部坐标即曲面局部坐标下进行,也可以在外部坐标即在笛卡尔坐标下进行。在微分几何中,对曲面的研究可以利用内禀(intrinsic)几何和外部(extrinsic)几何这两种互补也是基本的基本方法进行。但是量子力学的传统(其实也是整个现代物理例如广义相对论和规范场的研究传统)是只用内部坐标即曲面局部坐标,或者说从内禀的角度,对运动进行描述。那么是否可以也从三维笛卡尔坐标下进行?
这个问题的数学表述如下:对于一个二维正则曲面,需两个参量来对曲面进行曲面化(例如u, v),这样,曲面方程就是: r =( x (u , v ), y (u , v ), z (u , v))此时的厄密动量算符p_x , p_y ,p_x称为笛卡尔动量算符。它和曲面上的正则动量完全不同。这一正则动量一般来说不是有界算符(例如径向动量,而且由于坐标系的转换而改变,一直受到数学物理界的责难。
从2003年已经开始,我们开始研究笛卡尔动量算符。在有关研究的基础上,本论文证明了笛卡尔动量算符具有如下形式,其中Hn是一个几何不变量,称为平均曲率矢量场。这是一个不随坐标系的转换而改变的量。
进一步,体系Hamilton量似应为,。其实这是不行的,正确的哈密顿应为或:或:其中f_x , f_y ,f_z为u, v的非平凡函数。这是一类新的算符次序问题。本文还给出了这些函数满足的微分方程,并在一些具体例子中给出了方程的解。
在笛卡尔坐标下研究约束体系物理图象更清晰,经典对应更直接。本研究的结果加深了对量子力学与经典力学之间关系的理解。
The quantum motion constrained on quantum dots is currently a hot research topic in condensed matter physics. A very interesting issue is the geometrical shapes of quantum dots. There are toroidal surface and M?bius surface etc besides the typical sphere and spheroid. In classical mechanics, for a particle moving on the curved surface embedded in three dimension Cartesian coordinates, the local curved coordinates on the surface and the Cartesian coordinates play equal roles in the description of its classical motion. In the differential geometry, we can use two complementary and fundamental methods such as intrinsic geometry and extrinsic geometry as well. However, traditional quantum mechanics, also the tradition of the modern physics research such as general relativity and gauge field, uses the local coordinate system only. Then can we use the three dimension Cartesian coordinates to describe motion constrained on the two dimensional curved surfaces?
Expressing this problem in mathematical language, we have that two variables are needed to describe a two dimensional regular curved surface, for example (u , v ); and the surface equation is r =( x (u , v ), y (u , v ), z (u , v)), where the Hermitian momentum operators p_x , p_y ,p_x are the so called Cartesian momentum operators. They are entirely different from the canonical momentum operators on the curved surface. Generally speaking, these canonical momentum operators are not bounded operators, for example radial momentum operators , and their forms change with the coordinates transformations. This problem has been criticized by the mathematics and the physics circles.
The research of the Cartesian momentum operators began in 2003. This present work proves that the Cartesian momentum operators take the following forms: where Hn is a geometrical invariant, which is called mean curvature vector field. This is an invariant under the transformations of coordinates.
Furthermore, the Hamilton of the system seems to be but it is not the case. The correct Hamilton should be, where the f_x , f_y ,f_z are the nontrivial functions of u ,v . This is a new kind operator ordering problem. This paper also gives the differential equations determining these functions. Moreover, some explicit solutions are given for some concrete examples.
The physical picture with Cartesian operators is much clear and classical correspondence is straightforward. Our research casts a new insight into the understanding of the classical correspondence of quantum mechanics.
这个问题的数学表述如下:对于一个二维正则曲面,需两个参量来对曲面进行曲面化(例如u, v),这样,曲面方程就是: r =( x (u , v ), y (u , v ), z (u , v))此时的厄密动量算符p_x , p_y ,p_x称为笛卡尔动量算符。它和曲面上的正则动量完全不同。这一正则动量一般来说不是有界算符(例如径向动量,而且由于坐标系的转换而改变,一直受到数学物理界的责难。
从2003年已经开始,我们开始研究笛卡尔动量算符。在有关研究的基础上,本论文证明了笛卡尔动量算符具有如下形式,其中Hn是一个几何不变量,称为平均曲率矢量场。这是一个不随坐标系的转换而改变的量。
进一步,体系Hamilton量似应为,。其实这是不行的,正确的哈密顿应为或:或:其中f_x , f_y ,f_z为u, v的非平凡函数。这是一类新的算符次序问题。本文还给出了这些函数满足的微分方程,并在一些具体例子中给出了方程的解。
在笛卡尔坐标下研究约束体系物理图象更清晰,经典对应更直接。本研究的结果加深了对量子力学与经典力学之间关系的理解。
The quantum motion constrained on quantum dots is currently a hot research topic in condensed matter physics. A very interesting issue is the geometrical shapes of quantum dots. There are toroidal surface and M?bius surface etc besides the typical sphere and spheroid. In classical mechanics, for a particle moving on the curved surface embedded in three dimension Cartesian coordinates, the local curved coordinates on the surface and the Cartesian coordinates play equal roles in the description of its classical motion. In the differential geometry, we can use two complementary and fundamental methods such as intrinsic geometry and extrinsic geometry as well. However, traditional quantum mechanics, also the tradition of the modern physics research such as general relativity and gauge field, uses the local coordinate system only. Then can we use the three dimension Cartesian coordinates to describe motion constrained on the two dimensional curved surfaces?
Expressing this problem in mathematical language, we have that two variables are needed to describe a two dimensional regular curved surface, for example (u , v ); and the surface equation is r =( x (u , v ), y (u , v ), z (u , v)), where the Hermitian momentum operators p_x , p_y ,p_x are the so called Cartesian momentum operators. They are entirely different from the canonical momentum operators on the curved surface. Generally speaking, these canonical momentum operators are not bounded operators, for example radial momentum operators , and their forms change with the coordinates transformations. This problem has been criticized by the mathematics and the physics circles.
The research of the Cartesian momentum operators began in 2003. This present work proves that the Cartesian momentum operators take the following forms: where Hn is a geometrical invariant, which is called mean curvature vector field. This is an invariant under the transformations of coordinates.
Furthermore, the Hamilton of the system seems to be but it is not the case. The correct Hamilton should be, where the f_x , f_y ,f_z are the nontrivial functions of u ,v . This is a new kind operator ordering problem. This paper also gives the differential equations determining these functions. Moreover, some explicit solutions are given for some concrete examples.
The physical picture with Cartesian operators is much clear and classical correspondence is straightforward. Our research casts a new insight into the understanding of the classical correspondence of quantum mechanics.
引文
[1] Nielsen M A, Chuang I L. Quantum computation and quantum information. Cambridge: Cambridge University Press, 2000, 353-416
[2] Domingos J M. Foundations of physics. Cambridge: Cambridge University Press, 1984,147-148
[3] Borovitskaya E, Michael S S. Quantum dots. Singapore: World scientifics Publishing, 2002, 95-96
[4] Turton R. The quantum dot: a journey into the future of microelectronics. London: Oxford Univesity, 1996, 15-16
[5] 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, 82-85
[6] Bouwmeester D, Ekert A, Zeilinger A. The physics of quantum information. Berlin: Springer-Verlag, 2000, 132-140
[7] 张永德,吴盛俊,侯广,等. 量子信息论物理原理和某些进展. 武汉: 华中师范大学出版社, 2002, 23-30
[8] Hirvensalo M. Quantum computing. New York: Springer-verlag, 2001, 35-37
[9] 李承祖, 黄明球, 陈平形, 等. 量子通信和量子计算. 长沙: 国防科技大学出版社, 2001, 51-52
[10] Bennett C H, Brassard G. Event-ready-detectors bell experiment via entanglement swapping. Physical Review Letters, 1993, 71(26): 4287-4290
[11] 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. Physical Review Letters, 2005, 94(1): 107401-107402
[12] Wang L J, Kuzmich A, Dogariu A. Gain-asisted superluminal light propagation. Nature, 2000, 406(13): 277-278
[13] Huang C G, Zhang Y Z. Poynting vector, energy density, and energy velocity in an anomalous dispersion medium. Physical Review A, 2001, 65(14): 015802-015803
[14] Huang C G, Zhang Y Z. Propagation of a rectangular pulse in an anomalous dispersive medium. Communications in Theoretical Physics, 2002, 38(4): 224-226
[15] Scully M O. Enhancement of the index of refraction via quantum coherence.Phys Rev Lett, 1991, 67(14): 1855-1856
[16] Field J E, Hahn K H, Harris S E. Observation of electromagnetically induced transparency in collisionally broadened lead vapor. Phys Rev Lett, 1991, 67(22): 3062-3065
[17] Harris S E, Field J E, Kasapi A. Dispersive properties of electromagnetically induced transparency. Physical Review A, 1992, 46(1): 29-32
[18] Cantele G, Ninno D. Confined states in ellipsoidal quantum dots. Journal of Physics: Condensed Matter, 2000, 12(42): 9019-9036
[19] Cantele G, Ninno D, Iadonisi G. Shape effects on the one- and two-electron ground state in ellipsoidal quantum dots. Physical Review B, 2001, 64(12): 125325-125327
[20] Encinosa M, Mott L. Curvature-induced toroidal bound states. Physical Review A, 2003, 68(1): 014102-014106
[21] Gravesen J, Willatzen M. Eigenstates of m?bius nanostructures including curvature effects. Physical Review A, 2005, 72(10): 032108-032112
[22] Costa R C. Quantum mechanics of a constrained particle. Physics Review A, 1981, 23(3): 1982-1987
[23] 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
[24] Liu Q H, Hou J X, Xiao Y P, et al. Quantum motion on 2D surface of nonspherical topology. International Journal of Theoretical physics, 2004, 43(4): 1011-1017
[25] Xiao Y P, Lai M M, Hou J X, et al. A secondary operator ordering problem for a charged rigid planar rotator in the uniform magnetic field. Communications in Theoretical Physics, 2005, 44(7): 49-50
[26] Lai M M, Wang X, Xiao Y P, et al. Gauge transformation and constraint induced operator ordering for charged rigid planar rotator in uniform magnetic field. Communications in Theoretical Physics, 2006, 46(11): 843-844
[27] Wang X, Xiao Y P, Liu T G, et al. Quantum motion on 2D surface of spherical topology. International Journal of Theoretical physics, 2006, 45(12): 2464-2470
[28] Liu Q H. Universality of operator ordering in kinetic energy operator for particles moving on two dimensional surfaces. International of Theoretical Physics, 2006, 45(11): 2137-2142
[29] Liu Q H, Tong C L, Lai M M. Constraint induced mean curvature dependence of cartesian momentum operators. Journal of Physics A: Mathematical andTheoretical, 2007, 40(15): 4161-4168
[30] 黄克智, 薛明德, 陆明万. 张量分析. 北京: 清华大学出版社,2003, 37-40
[31] Podolsky B. Quantum-mechanically correct form of hamiltonian function for conservative systems. Physical Review,1928, 32(5): 812-816
[32] Kleinert H. Path integrals in quantum mechanics. Singapore: World Scientific, 1990, 186-187
[33] Bohm D. Quantum Theory. New Jersey: Prentice-Hall, 1951, 36-48
[34] 王竹溪, 郭敦仁. 特殊函数概论. 北京: 科学出版社, 1979, 17-25
[35] 肖艳萍. 约束引起的算符次序问题及其解决:[湖南大学硕士学位论文].长沙:湖南大学, 2005, 10-15
[36] 梅向明, 黄敬之. 微分几何. 北京: 高等教育出版社, 1988, 60-62
[37] 张永德. 量子力学. 北京: 科学出版社, 2003, 15-16
[38] Ou-yang Z C, Liu J X , Xie Y Z. Geometrical methods in the elastic theory of membraines in liquid crystal phases. Singapore: World Scientific, 1999, 51-52
[39] 童纯立. 约束引起的算符次序问题的微分几何处理: [湖南大学本科学位论文]. 长沙: 湖南大学, 2006, 13-14
[40] Liu Q H. The classical limit of quantum mechanics and the fejér sum of the fourier series expansion of a classical quantity. Journal of Physics A: Mathematical and General, 1999, 32(4): 57-62
[41] Liu Q H, Wang X, Qi W H, et al. The Fejér average and the mean value of a quantity in a quasiclassical wave packet. Journal of Mathmetical physics, 2002 , 43(6): 170-181
[42] Liu Q H, Hu B. The hydrogen atom's quantum-to-classical correspondence in heisenberg's correspondence principle. Journal of Physics A: Mathematical and General, 2001, 34(28):5713-5719
[43] Liu Q H, Wang X, Yu Y B. Total current operator and its classical correspondence for particles bounded in central force fields. International Journal of Theoretical Physics, 2007, 46(3): 424-429
[2] Domingos J M. Foundations of physics. Cambridge: Cambridge University Press, 1984,147-148
[3] Borovitskaya E, Michael S S. Quantum dots. Singapore: World scientifics Publishing, 2002, 95-96
[4] Turton R. The quantum dot: a journey into the future of microelectronics. London: Oxford Univesity, 1996, 15-16
[5] 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, 82-85
[6] Bouwmeester D, Ekert A, Zeilinger A. The physics of quantum information. Berlin: Springer-Verlag, 2000, 132-140
[7] 张永德,吴盛俊,侯广,等. 量子信息论物理原理和某些进展. 武汉: 华中师范大学出版社, 2002, 23-30
[8] Hirvensalo M. Quantum computing. New York: Springer-verlag, 2001, 35-37
[9] 李承祖, 黄明球, 陈平形, 等. 量子通信和量子计算. 长沙: 国防科技大学出版社, 2001, 51-52
[10] Bennett C H, Brassard G. Event-ready-detectors bell experiment via entanglement swapping. Physical Review Letters, 1993, 71(26): 4287-4290
[11] 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. Physical Review Letters, 2005, 94(1): 107401-107402
[12] Wang L J, Kuzmich A, Dogariu A. Gain-asisted superluminal light propagation. Nature, 2000, 406(13): 277-278
[13] Huang C G, Zhang Y Z. Poynting vector, energy density, and energy velocity in an anomalous dispersion medium. Physical Review A, 2001, 65(14): 015802-015803
[14] Huang C G, Zhang Y Z. Propagation of a rectangular pulse in an anomalous dispersive medium. Communications in Theoretical Physics, 2002, 38(4): 224-226
[15] Scully M O. Enhancement of the index of refraction via quantum coherence.Phys Rev Lett, 1991, 67(14): 1855-1856
[16] Field J E, Hahn K H, Harris S E. Observation of electromagnetically induced transparency in collisionally broadened lead vapor. Phys Rev Lett, 1991, 67(22): 3062-3065
[17] Harris S E, Field J E, Kasapi A. Dispersive properties of electromagnetically induced transparency. Physical Review A, 1992, 46(1): 29-32
[18] Cantele G, Ninno D. Confined states in ellipsoidal quantum dots. Journal of Physics: Condensed Matter, 2000, 12(42): 9019-9036
[19] Cantele G, Ninno D, Iadonisi G. Shape effects on the one- and two-electron ground state in ellipsoidal quantum dots. Physical Review B, 2001, 64(12): 125325-125327
[20] Encinosa M, Mott L. Curvature-induced toroidal bound states. Physical Review A, 2003, 68(1): 014102-014106
[21] Gravesen J, Willatzen M. Eigenstates of m?bius nanostructures including curvature effects. Physical Review A, 2005, 72(10): 032108-032112
[22] Costa R C. Quantum mechanics of a constrained particle. Physics Review A, 1981, 23(3): 1982-1987
[23] 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
[24] Liu Q H, Hou J X, Xiao Y P, et al. Quantum motion on 2D surface of nonspherical topology. International Journal of Theoretical physics, 2004, 43(4): 1011-1017
[25] Xiao Y P, Lai M M, Hou J X, et al. A secondary operator ordering problem for a charged rigid planar rotator in the uniform magnetic field. Communications in Theoretical Physics, 2005, 44(7): 49-50
[26] Lai M M, Wang X, Xiao Y P, et al. Gauge transformation and constraint induced operator ordering for charged rigid planar rotator in uniform magnetic field. Communications in Theoretical Physics, 2006, 46(11): 843-844
[27] Wang X, Xiao Y P, Liu T G, et al. Quantum motion on 2D surface of spherical topology. International Journal of Theoretical physics, 2006, 45(12): 2464-2470
[28] Liu Q H. Universality of operator ordering in kinetic energy operator for particles moving on two dimensional surfaces. International of Theoretical Physics, 2006, 45(11): 2137-2142
[29] Liu Q H, Tong C L, Lai M M. Constraint induced mean curvature dependence of cartesian momentum operators. Journal of Physics A: Mathematical andTheoretical, 2007, 40(15): 4161-4168
[30] 黄克智, 薛明德, 陆明万. 张量分析. 北京: 清华大学出版社,2003, 37-40
[31] Podolsky B. Quantum-mechanically correct form of hamiltonian function for conservative systems. Physical Review,1928, 32(5): 812-816
[32] Kleinert H. Path integrals in quantum mechanics. Singapore: World Scientific, 1990, 186-187
[33] Bohm D. Quantum Theory. New Jersey: Prentice-Hall, 1951, 36-48
[34] 王竹溪, 郭敦仁. 特殊函数概论. 北京: 科学出版社, 1979, 17-25
[35] 肖艳萍. 约束引起的算符次序问题及其解决:[湖南大学硕士学位论文].长沙:湖南大学, 2005, 10-15
[36] 梅向明, 黄敬之. 微分几何. 北京: 高等教育出版社, 1988, 60-62
[37] 张永德. 量子力学. 北京: 科学出版社, 2003, 15-16
[38] Ou-yang Z C, Liu J X , Xie Y Z. Geometrical methods in the elastic theory of membraines in liquid crystal phases. Singapore: World Scientific, 1999, 51-52
[39] 童纯立. 约束引起的算符次序问题的微分几何处理: [湖南大学本科学位论文]. 长沙: 湖南大学, 2006, 13-14
[40] Liu Q H. The classical limit of quantum mechanics and the fejér sum of the fourier series expansion of a classical quantity. Journal of Physics A: Mathematical and General, 1999, 32(4): 57-62
[41] Liu Q H, Wang X, Qi W H, et al. The Fejér average and the mean value of a quantity in a quasiclassical wave packet. Journal of Mathmetical physics, 2002 , 43(6): 170-181
[42] Liu Q H, Hu B. The hydrogen atom's quantum-to-classical correspondence in heisenberg's correspondence principle. Journal of Physics A: Mathematical and General, 2001, 34(28):5713-5719
[43] Liu Q H, Wang X, Yu Y B. Total current operator and its classical correspondence for particles bounded in central force fields. International Journal of Theoretical Physics, 2007, 46(3): 424-429