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arXiv:1803.05509 · Modern Physics Letters A

Geometric momentum and angular momentum for charge-monopole system

S. F. Xiao, Q. H. Liu

Published Jul 17, 2018• 1 citation
arXivHTMLPDF

Abstract

For a charge-monopole pair, we have another definition of the orbital angular momentum, and the transverse part of the momentum including the vector potential turns out to be the so-called geometric momentum that is under intensive study recently. For the charge on the spherical surface with the monopole at the origin, the commutation relations between all components of both the geometric momentum and the orbital angular momentum satisfy the so(3,[Formula: see text]1) algebra. With construction of the geometrically infinitesimal displacement operator based on the geometric momentum, the so(3,[Formula: see text]1) algebra implies the Aharonov–Bohm phase shift. The related problems such as charge and flux quantization are also addressed.

DOI
10.1142/s0217732318501250

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For a charge-monopole pair, we have another definition of the orbital angular momentum, and the transverse part of the momentum including the vector potential turns out to be the so-called geometric momentum that is under intensive study recently. For the charge on the spherical surface with the monopole at the origin, the commutation relations between all components of both the geometric momentum and the orbital angular momentum satisfy the so(3,[Formula: see text]1) algebra. With construction of the geometrically infinitesimal displacement operator based on the geometric momentum, the so(3,[Formula: see text]1) algebra implies the Aharonov–Bohm phase shift. The related problems such as charge and flux quantization are also addressed.
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 pi{p}_{i}pi​ (i=1,2,3)(i=1,2,3)(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 pi{p}_{i}pi​ 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.
The orbital angular momentum operator is defined and its commutation relations with the position and momentum operators, and generally with vector operators, are obtained; the relationship between the square of the momentum operator and the square of the orbital angular momentum is derived; the spectrum of the square and z-component of the orbital angular momentum is obtained by solving the Schröedinger equation near the origin; the radial equation is derived and the spherical harmonics are obtained as solutions of the associated Legendre equation.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
Recently the physical mechanism for geometric phase in optics has been elucidated in terms of the angular momentum holonomy proposed in 1992. Aharonov and Kaufherr (PRL, 92, 070404, 2004) revisit the Aharonov-Bohm effect, and propose non-local exchange of a conserved, gauge invariant quantity that changes the modular momentum of the particle that is responsible for the AB phase shift. We suggest that the net angular momentum shifts proposed for GP may be analogous to the shift in the modulus momentum for the AB effect. At a single photon or electron level such non-trivial, geometric effects seem to hint at a new physics.
The angular momentum plays an important role in quantum mechanics. It is no longer an ordinary vector, instead it is a vector operator, whose three components do not commute with each other. In quantum mechanics, the angular momentum becomes a vector operator. A general angular momentum is defined as a vector operator whose components are Hermitian operators satisfying the commutation relations similar to those for the orbital angular momentum. In order to obtain the eigenvalues and eigenfunctions, it is more convenient to express the orbital angular momentum operators in spherical polar coordinates. Spin angular momentum is an intrinsic property of elementary particles. In quantum computing, a qubit or quantum bit is a quantum version of the classical binary bit physically realized in a two-state quantum-mechanical system.
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.