径向动量的等效算符及其量子测量
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摘要
提出了一种在正交曲线坐标系分解动量的新方法,进而给出了球坐标下径向动量算符的一种等效表示.尽管径向动量算符本身不能测量,这个等效算符却能测量,从而解决了径向动量的非自伴性但是具有不确定度之间的矛盾.以基态氢原子为例,给出了等效径向动量的值的分布.
A new decomposition of the momentum operator in curvilinear coordinates is proposed, from which an equivalent operator of radial operator is resulted. Though the radial operator itself is non-self-adjoint and is not measurable in physics, the equivalent operator is in fact the difference of two self-adjoint operators, so is measurable. Thus, a superficial contradiction between the non-self-adjointness and existence of the uncertainty for the radial momentum operator is resolved. What is more, the Hydrogen atom ground state is used to show how to measure the equivalent operator.
A new decomposition of the momentum operator in curvilinear coordinates is proposed, from which an equivalent operator of radial operator is resulted. Though the radial operator itself is non-self-adjoint and is not measurable in physics, the equivalent operator is in fact the difference of two self-adjoint operators, so is measurable. Thus, a superficial contradiction between the non-self-adjointness and existence of the uncertainty for the radial momentum operator is resolved. What is more, the Hydrogen atom ground state is used to show how to measure the equivalent operator.
引文
[1] Dirac P A M.The Principles of Quantum Mechanics[M].4th.Oxford,Oxford University Press,1967:152-153.
[2] Domingos J M,Caldeira M.Self-adjointness of momentum operators in generalized coordinates[J].Found Phys,14(1984)147.
[3] A 梅西亚.量子力学(I)[M].苏汝铿,等译.北京:科学出版社,1986:193.
[4] 罗凌霄.氢原子和类氢离子的等式型径向动量-位置不确定关系[J].大学物理.2006,22(2):122-124.
[5] Liu Q H,Xiao S F.A self-adjoint decomposition of the radial momentum operator[J].International Journal of Geometric Methods in Modern Physics,2015,12(3):1550028-1550035.
[6] 王竹溪,郭敦仁.特殊函数概论[M].北京:北京大学出版社,2000:650.
[7] W Pauli.General principles of quantum mechanics[M].Berlin:Springer-Verlag,1980.
[8] 陈维桓.微分几何[M].北京:北京大学出版社,1990.
[9] Liu Q H,Tang L H,Xun D M.Geometric momentum:The proper momentum for a free particle on a two-dimensional sphere[J].Physical Review A,2011,84(4):042101.
[10] Liu Q H.Geometric Momentum and a Probe of Embeding Effects[J].Journal of the Physical Society of Japan,2013,82(10):104002-104009.
[11] Liu Q H.Geometric momentum for a particle constrained on a curved hypersurface[J] Journal of Mathematical Physics.2013,54(12):122113.
[12] Lian D K,Hu L D,Liu Q H.Geometric Potential and Dirac Quantization[J].Annalen Physik(Berlin),2018,530:1700415-1700418.
[13] Ron S,Patrick U,Hartmut B,et al.Curvature-induced geometric momenta:the origin of waveguide dispersion of surface plasmons on metallic wires[J].Optics Express,2015,23(9):12174-12188.
[14] 刘全慧,张梦南,肖世发,等.三维各向同性谐振子的几何动量分布[J].物理学报,2019,68(1):010301
[2] Domingos J M,Caldeira M.Self-adjointness of momentum operators in generalized coordinates[J].Found Phys,14(1984)147.
[3] A 梅西亚.量子力学(I)[M].苏汝铿,等译.北京:科学出版社,1986:193.
[4] 罗凌霄.氢原子和类氢离子的等式型径向动量-位置不确定关系[J].大学物理.2006,22(2):122-124.
[5] Liu Q H,Xiao S F.A self-adjoint decomposition of the radial momentum operator[J].International Journal of Geometric Methods in Modern Physics,2015,12(3):1550028-1550035.
[6] 王竹溪,郭敦仁.特殊函数概论[M].北京:北京大学出版社,2000:650.
[7] W Pauli.General principles of quantum mechanics[M].Berlin:Springer-Verlag,1980.
[8] 陈维桓.微分几何[M].北京:北京大学出版社,1990.
[9] Liu Q H,Tang L H,Xun D M.Geometric momentum:The proper momentum for a free particle on a two-dimensional sphere[J].Physical Review A,2011,84(4):042101.
[10] Liu Q H.Geometric Momentum and a Probe of Embeding Effects[J].Journal of the Physical Society of Japan,2013,82(10):104002-104009.
[11] Liu Q H.Geometric momentum for a particle constrained on a curved hypersurface[J] Journal of Mathematical Physics.2013,54(12):122113.
[12] Lian D K,Hu L D,Liu Q H.Geometric Potential and Dirac Quantization[J].Annalen Physik(Berlin),2018,530:1700415-1700418.
[13] Ron S,Patrick U,Hartmut B,et al.Curvature-induced geometric momenta:the origin of waveguide dispersion of surface plasmons on metallic wires[J].Optics Express,2015,23(9):12174-12188.
[14] 刘全慧,张梦南,肖世发,等.三维各向同性谐振子的几何动量分布[J].物理学报,2019,68(1):010301