Geometric momentum distribution for three-dimensional isotropic hormonic oscillator

Abstract

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<em>ħ</em>▽= <strong><em>p</em></strong> _<em>Σ</em> + <strong><em>p</em></strong> _<em>n</em> where-i<em>ħ</em>▽ can be usual momentum operator in Cartesian coordinates, and <strong><em>p</em></strong> _<em>Σ</em> is the momentum component on the hypersurface which turns out to be the geometric momentum, and <strong><em>p</em></strong> _<em>n</em> is the momentum component along the radial direction, we have a nontrivial definition of radial momentum as <strong><em>p</em></strong> _<em>n</em> ≡-i<em>ħ</em>▽- <strong><em>p</em></strong> _<em>Σ</em>. Once-i<em>ħ</em>▽ and <strong><em>p</em></strong> _<em>Σ</em> are measurable, <strong><em>p</em></strong> _<em>n</em> 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.