三维各向同性谐振子的几何动量分布
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摘要
尽管几何动量最初的引入是为了描述超面上的运动粒子的动量,却不需要限制在真实的曲面上.如果一个曲线坐标系包含了超面族和超面上的法向矢量作为一个坐标轴的单位矢量,几何动量可以定义在超面族上,并参与构造对易力学量完全集.在三维各向同性谐振子中,采用球坐标描述,存在等效球面,并在球面族上建立对易力学量完全集.因此,三维各向同性谐振子同时具有动量和几何动量分布.这两个动量的差,可以定义为径向动量,从而使得径向动量可以测量.那么,通过几何动量,可以显示出狄拉克引进的径向动量的物理意义,而不是一直认为的那样完全不具有观测意义.
The geometric momentum was originally introduced for defining the momentum of particle constrained on a hypersurface, but it is in fact not necessarily defined on a curved surface only. If a coordinate system contains a family of hypersurfaces and a normal vector on hypersurface used as a unit vector, the geometric momentum can be defined on the family of hypersurfaces and can be used to determine a complete set of commuting observables. For instance,the spherical polar coordinate system is such a kind of coordinate, in which for a given value of radial position, the spherical surface is a hypersurface. It is well-known that any vector in the space can be decomposed into components along each axis of the spherical polar coordinates, but the geometric momentum has a different decomposition, for it requires a projection of the momentum on the hypersurface, and then needs to decompose the projection into the Cartesian coordinates of the original space where the whole spherical coordinates are defined. Explicitly, with a relation-i = p_Σ+ p_n where-i can be usual momentum operator in Cartesian coordinates, and pΣis the momentum component on the hypersurface which turns out to be the geometric momentum, and pnis the momentum component along the radial direction, we have a nontrivial definition of radial momentum as pn≡-i-p_Σ. Once-i and p_Σ are measurable, pnis then indirectly measurable. The three-dimensional isotropic harmonic oscillator can be described in both the Cartesian and the spherical polar coordinates, whose quantum states thus can be examined in terms of both momentum and geometric momentum distributions. The distributions of the radial momentum are explicitly given for some states. The radial momentum operator that was introduced by Dirac has clear physical significance, in contrast to widely spreading belief that it is not measurable due to its non-self-adjoint.
The geometric momentum was originally introduced for defining the momentum of particle constrained on a hypersurface, but it is in fact not necessarily defined on a curved surface only. If a coordinate system contains a family of hypersurfaces and a normal vector on hypersurface used as a unit vector, the geometric momentum can be defined on the family of hypersurfaces and can be used to determine a complete set of commuting observables. For instance,the spherical polar coordinate system is such a kind of coordinate, in which for a given value of radial position, the spherical surface is a hypersurface. It is well-known that any vector in the space can be decomposed into components along each axis of the spherical polar coordinates, but the geometric momentum has a different decomposition, for it requires a projection of the momentum on the hypersurface, and then needs to decompose the projection into the Cartesian coordinates of the original space where the whole spherical coordinates are defined. Explicitly, with a relation-i = p_Σ+ p_n where-i can be usual momentum operator in Cartesian coordinates, and pΣis the momentum component on the hypersurface which turns out to be the geometric momentum, and pnis the momentum component along the radial direction, we have a nontrivial definition of radial momentum as pn≡-i-p_Σ. Once-i and p_Σ are measurable, pnis then indirectly measurable. The three-dimensional isotropic harmonic oscillator can be described in both the Cartesian and the spherical polar coordinates, whose quantum states thus can be examined in terms of both momentum and geometric momentum distributions. The distributions of the radial momentum are explicitly given for some states. The radial momentum operator that was introduced by Dirac has clear physical significance, in contrast to widely spreading belief that it is not measurable due to its non-self-adjoint.
引文
[1]Liu Q H,Tang L H,Xun D M 2011 Phys.Rev.A 84042101
[2]Spittel R,Uebel P,Bartelt H,Schmidt M A 2015 Opt.Express 23 12174
[3]Liu Q H 2013 J.Phys.Soc.Jpn.82 104002
[4]Xun D M,Liu Q H 2013 Int.J.Geom.Methods Mod.Phys.10 1220031
[5]Xun D M,Liu Q H,Zhu X M 2013 Ann.Phys.(N.Y.)338 123
[6]Xun D M,Liu Q H 2014 Ann.Phys.(N.Y.)341 132
[7]Zhang Z S,Xiao S F,Xun D M,Liu Q H 2015 Commun.Theor.Phys.63 19
[8]Xun D M,Ouyang T,Tan R R,Liu H X 2015 Acta Phys.Sin.64 240305(in Chinese)[寻大毛,欧阳涛,谈荣日,刘慧宣2015物理学报64 240305]
[9]Wang Y L,Du L,Xu C T,Liu X J,Zong H S 2014 Phys.Rev.A 90 042117
[10]Wang Y L,Jiang H,Zong H S 2017 Phys.Rev.A 96022116
[11]Lian D K,Hu L D,Liu Q H 2018 Ann.Phys.5301700415
[12]Liu Q H 2013 J.Math.Phys.54 122113
[13]Dirac P A M 1967 The Principles of Quantum Mechanics(4th edition)(Oxford:Oxford University Press)p114,p153
[14]Robinson P D,Hirschfelder J O 1963 J.Math.Phys.4338
[15]Arthurs A M 1968 Proc.Natl.Acad.Sci.USA 60 1105
[16]Domingos J M,Caldeira M H 1984 Found.Phys.14 147
[17]Liu Q H,Xiao S F 2015 Int.J.Geom.Methods Mod.Phys.12 1550028
[18]Xiao S F,Liu Q H 2018 Mod.Phys.Lett.A 33 1850125
[19]Liu Q H 2014 Phys.Lett.A 378 785
[20]Yang C N 1977 Ann.NY Acad.Sci.294 86
[21]Zeng J Y 2014 Quantum Mechanics(Vol.1)(Fifth edition)(Beijing:Science Press)pp206-211(in Chinese)[曾谨言2014量子力学(上卷)(第五版)(北京:科学出版社)第206-211页]
[2]Spittel R,Uebel P,Bartelt H,Schmidt M A 2015 Opt.Express 23 12174
[3]Liu Q H 2013 J.Phys.Soc.Jpn.82 104002
[4]Xun D M,Liu Q H 2013 Int.J.Geom.Methods Mod.Phys.10 1220031
[5]Xun D M,Liu Q H,Zhu X M 2013 Ann.Phys.(N.Y.)338 123
[6]Xun D M,Liu Q H 2014 Ann.Phys.(N.Y.)341 132
[7]Zhang Z S,Xiao S F,Xun D M,Liu Q H 2015 Commun.Theor.Phys.63 19
[8]Xun D M,Ouyang T,Tan R R,Liu H X 2015 Acta Phys.Sin.64 240305(in Chinese)[寻大毛,欧阳涛,谈荣日,刘慧宣2015物理学报64 240305]
[9]Wang Y L,Du L,Xu C T,Liu X J,Zong H S 2014 Phys.Rev.A 90 042117
[10]Wang Y L,Jiang H,Zong H S 2017 Phys.Rev.A 96022116
[11]Lian D K,Hu L D,Liu Q H 2018 Ann.Phys.5301700415
[12]Liu Q H 2013 J.Math.Phys.54 122113
[13]Dirac P A M 1967 The Principles of Quantum Mechanics(4th edition)(Oxford:Oxford University Press)p114,p153
[14]Robinson P D,Hirschfelder J O 1963 J.Math.Phys.4338
[15]Arthurs A M 1968 Proc.Natl.Acad.Sci.USA 60 1105
[16]Domingos J M,Caldeira M H 1984 Found.Phys.14 147
[17]Liu Q H,Xiao S F 2015 Int.J.Geom.Methods Mod.Phys.12 1550028
[18]Xiao S F,Liu Q H 2018 Mod.Phys.Lett.A 33 1850125
[19]Liu Q H 2014 Phys.Lett.A 378 785
[20]Yang C N 1977 Ann.NY Acad.Sci.294 86
[21]Zeng J Y 2014 Quantum Mechanics(Vol.1)(Fifth edition)(Beijing:Science Press)pp206-211(in Chinese)[曾谨言2014量子力学(上卷)(第五版)(北京:科学出版社)第206-211页]