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Acta Physica Sinica

Geometric momentum distribution for three-dimensional isotropic hormonic oscillator

Quanhui Liu, Zhang Meng-Nan, Shi-Fa Xiao, Damao Xun

Published Jan 1, 2019• 0 citations

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-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.

DOI: 10.7498/aps.68.20181634

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•Sep 18, 2017•0 citations

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.

As a manifold is embedded into a higher dimensional Euclidean space, quantum mechanics gives various embedding quantities. In the present study, two embedding quantities for a two-dimensional curved surface in the three-dimensional flat space, the geometric momentum and the geometric potential, are derived in a unified manner. For a particle moving on a two-dimensional sphere or a free rotation of a spherical top, the projections of both the geometric momentum

\mathbf{p}

and the angular momentum

\mathbf{L}

onto a certain Cartesian axis form a complete set of commuting observables as

[p_{i},L_{i}]=0

(

i=1,2,3

), thus constituting a dynamical

(p_{i},L_{i})

representation for the states on the two-dimensional spherical surface. The geometric momentum distribution of the states represented by spherical harmonics is successfully obtained, and this distribution for a homonuclear diatomic molecule seems within the resolution power of present momentum spectrometers and can be measured to probe the embedding effect.

In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum, and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum ${p}_{\ensuremath{\theta}}$ breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as

{p}_{i}

(i=1,2,3)

are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between

{p}_{i}

generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.

As a nonrelativistic particle constrained to remain on an

N-1

(

N\geq 2

) dimensional hypersurface (embedded in an

N

dimensional Euclidean space), two different components

p_{i}

and

p_{j}

(

i,j=1,2,3,...N

) of the Cartesian momentum are not commuting with each other in the sense of the Dirac bracket

[p_{i},p_{j}]_{D}\neq 0

. The noncommutativity on a local point of the hypersurface is examined and results show that it is arisen from the local curvature of the surface. In quantum mechanics, with the infinitesimal displacement operator constructed via the momentum, and letting the infinitesimal displacement operator acting on a quantum state along a small circle on the hypersurface, we obtain a rotation operator, and the amount of the rotation is an angle anholonomy.

As a submanifold is embedded into higher dimensional flat space, quantum mechanics gives various embedding quantities, e.g., the geometric momentum and geometric potential, etc. For a particle moving on a two-dimensional sphere or a free rotation of a spherical top, the projections of the geometric momentum p and the angular momentum L onto a certain Cartesian axis form a complete set of commuting observables as [p_{i},L_{i}]=0 (i=1,2,3). We have therefore a (p_{i},L_{i}) representation for the states on the two-dimensional spherical surface. The geometric momentum distribution of the ground states for a freely rotating rigid rotor seems within the resolution power of present momentum spectrometer and can be measured to probe the embedding effect.

Abstract Geometric momentum is the appropriate momentum for a particle constrained to move on a curved surface, which depends on the extrinsic curvature and leads to observable effects, and curvature-induced quantum potentials appear for a nonrelativistic free particle on the surface. In the context of multi-component quantum states, the geometric momentum should be rewritten as a generally covariant geometric momentum, which contains an additional term defined as the gauge potential. For a Dirac fermion constrained on a two-dimensional hypersurface, we derive the generally covariant geometric momentum, and demonstrate that no curvature-induced geometric potentials arise on a pseudosphere or a helical surface. The dynamical quantization conditions are verified to be effective in dealing with constrained systems on hypersurfaces, enabling the derivation of both the generally convariant geometric momentum and the geometric potential for a spin particle constrained on parametrically defined surfaces.

As a nonrelativistic particle constrained to remain on an

N-1

(

N\geq 2

) dimensional hypersurface embedded in an

N

dimensional Euclidean space, two different components

p_{i}

and

p_{j}

(

i,j=1,2,3,...N

) of the Cartesian momentum of the particle are not commuting with each other. In fact, the so-called \textit{fundamental quantum conditions} $[p_{i},p_{j}]\left( \neq 0\right) $ depend on products of positions and momenta in uncontrollable ways, thus their physical consequences are little known. The local analysis technique is utilized to crack this long-standing problem. The first finding is that this noncommutativity on a \textit{local point }of the hypersurface turns out to be transparent and the quantum conditions are then operator-ordering free and dependent on the local curvature of the surface. The second finding is that, for a small circle lying on each of the \emph{% tangential planes} covering the \textit{local point}, the noncommutativity leads to a rotation operator and the amount of the rotation is an angle anholonomy; and along each of the \emph{normal sectional curves} centering the given point the noncommutativity leads to a translation plus an additional rotation and the amount of the rotation is one half of the tangential angle change of the arc.