Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

Abstract

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 θ 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.

Geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space

“…The definition of Eq. (13) without the surface component is different from the original geometric momentum [11,12] which is identical to our effective momentum. Our effective momentum consists of surface and geometric components, that is…”

“…The effective potential has been realized experimentally in photonic topological crystal [7], and the geometric influence on the quantum transport of two-dimensional (2D) curved materials has been investigated [8][9][10]. Another important result is the geometric influence on momentum [11,12] that has been observed governing the propagation of surface plasmon on metallic wires [13]. For full generality, as a curved surface is embedded in a higherdimensional (HD) Euclidean space, a novel geometrically induced gauge potential is present only when the space of normal states is degenerate [14].…”

“…In the same case, the momentum operator P is (12) where e 1 , e 2 and e n are three unit vectors with respect to q 1 , q 2 and q 3 , respectively. In terms of Eqs.…”

Dirac Equation in the Presence of Hartmann and Ring-Shaped Oscillator Potentials

“…The relativistic linear interaction, which is called the relativistic oscillator due to the similarity with the nonrelativistic harmonic oscillator, has been subject of many successful theoretical studies. Such a space has interesting property and algebra; for example, there are some articles in which a free particle has been studied in different situations; Dirac oscillator system that is initiated by a relativistic fermion is subjected to linear vector potential [23][24][25]. In this article, for solving Dirac equation with Hartmann and RingShaped Oscillator Potentials in three dimensions, equality of scalar and vector potentials can constitute a couple of differential equations for the spinor components [26,27].…”

Quantum particle confined to a thin-layer volume: Non-uniform convergence toward the curved surface

“…In the limited case q 3 = 0, the equation (31) is completely equivalent to the result in [2] when the electromagnetic field is vanished. The presence of the geometrical potential newly defines the geometric momentum [24] to replace the usual momentum. It is interesting to further study the modification induced by the thickness of surface to the geometric momentum.…”