Geometric momentum for a particle constrained on a curved hypersurface

Abstract

A strengthened canonical quantization scheme for the constrained motion on a curved hypersurface is proposed with introduction of the second category of fundamental commutation relations between Hamiltonian and positions/momenta, whereas those between positions and moments are categorized into the first. As an $N-1$ ($N\geq2$) dimensional hypersurface is embedded in an N dimensional Euclidean space, we obtain the proper momentum that depends on the mean curvature. For the surface is the spherical one, a long-standing problem on the form of the geometric potential is resolved in a lucid and unambiguous manner, which turns out to be identical to that given by the so-called confining potential technique. In addition, a new dynamical group SO(N,1) symmetry for the motion on the sphere is demonstrated.Comment: 5 pages, no figur

The centripetal force law and the equation of motion for a particle on a curved hypersurface

“…This curvature-induced potential has been confirmed by experiments [21,22] and may play some role in understanding of our present universe [2]. An illuminative exploration is to compare both sides of Eq.…”

“…In quantum mechanics, no momentum p = {p x , p y , p z } could be taken to substitute into the term ( p 2 x + p 2 y + p 2 z )/2μ so as to get the satisfactory quantum-mechanical operator 2 is the geometric potential obtained by the confining potential technique [20], with k being the extrinsic curvature tensor [3,5]. This curvature-induced potential has been confirmed by experiments [21,22] and may play some role in understanding of our present universe [2].…”

“…where the vector operator ∇ S ≡ r μ ∂ μ is the gradient operator defined on the surface, and −∇ S · n = M is the mean curvature [2]. In contrast, the operator O[p · ∇n · p] in (18) gives us many choices as regards the dependence on the components of position and momentum [16], because we are free to choose not only independent coordinates but also momenta; for we have two constrained conditions f (r) = 0 and n · p = 0.…”

“…The motion of a particle on a curved hypersurface is an exactly solvable model to examine various problems such as higher-dimensional gravity [1], the dark energy/matter problem [2], the quantization of constrained motions for a nonrelativistic particle [3][4][5][6][7] and for a relativistic fermion [8,9], and many curvature-induced effects in lower-dimensional systems and nanostructures [10][11][12][13][14][15], etc. We are familiar with both the geodesic equation from the intrinsically curved surface and the equation of motion from the extrinsically Euclidean space [5,6], but no relationship in between has been seriously explored.…”

Heisenberg equation for a nonrelativistic particle on a hypersurface: From the centripetal force to a curvature induced force

“…Substitution of F j (15), G j (16) and [p j , 2 4µ n i,l 2 ] (10) into Eq. (9) directly leads to the result (3).…”