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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-iħ▽= p Σ + p n where-iħ▽ can be usual momentum operator in Cartesian coordinates, and p Σ is the momentum component on the hypersurface which turns out to be the geometric momentum, and p n is the momentum component along the radial direction, we have a nontrivial definition of radial momentum as p n ≡-iħ▽- p Σ. Once-iħ▽ and p Σ are measurable, p n is 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.
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Keywords:
- angular momentum /
- harmonic oscillator /
- geometric momentum
[1] Liu Q H, Tang L H, Xun D M 2011 Phys. Rev. A 84 042101
[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] Wang Y L, Du L, Xu C T, Liu X J, Zong H S 2014 Phys. Rev. A 90 042117
[9] Wang Y L, Jiang H, Zong H S 2017 Phys. Rev. A 96 022116
[10] Lian D K, Hu L D, Liu Q H 2018 Ann. Phys. 530 1700415
[11] Liu Q H 2013 J. Math. Phys. 54 122113
[12] Dirac P A M 1967 The Principles of Quantum Mechanics (4th edition) (Oxford: Oxford University Press) p114, p153
[13] Robinson P D, Hirschfelder J O 1963 J. Math. Phys. 4 338
[14] Arthurs A M 1968 Proc. Natl. Acad. Sci. USA 60 1105
[15] Domingos J M, Caldeira M H 1984 Found. Phys. 14 147
[16] Liu Q H, Xiao S F 2015 Int. J. Geom. Methods Mod. Phys. 12 1550028
[17] Xiao S F, Liu Q H 2018 Mod. Phys. Lett. A 33 1850125
[18] Liu Q H 2014 Phys. Lett. A 378 785
[19] Yang C N 1977 Ann. NY Acad. Sci. 294 86
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[1] Liu Q H, Tang L H, Xun D M 2011 Phys. Rev. A 84 042101
[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] Wang Y L, Du L, Xu C T, Liu X J, Zong H S 2014 Phys. Rev. A 90 042117
[9] Wang Y L, Jiang H, Zong H S 2017 Phys. Rev. A 96 022116
[10] Lian D K, Hu L D, Liu Q H 2018 Ann. Phys. 530 1700415
[11] Liu Q H 2013 J. Math. Phys. 54 122113
[12] Dirac P A M 1967 The Principles of Quantum Mechanics (4th edition) (Oxford: Oxford University Press) p114, p153
[13] Robinson P D, Hirschfelder J O 1963 J. Math. Phys. 4 338
[14] Arthurs A M 1968 Proc. Natl. Acad. Sci. USA 60 1105
[15] Domingos J M, Caldeira M H 1984 Found. Phys. 14 147
[16] Liu Q H, Xiao S F 2015 Int. J. Geom. Methods Mod. Phys. 12 1550028
[17] Xiao S F, Liu Q H 2018 Mod. Phys. Lett. A 33 1850125
[18] Liu Q H 2014 Phys. Lett. A 378 785
[19] Yang C N 1977 Ann. NY Acad. Sci. 294 86
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