Geometric Momentum for a Particle on a Curved Surface

TL;DR

This paper introduces the concept of geometric momentum for particles on curved surfaces, derived via a confining procedure, and demonstrates its compatibility with quantum theory and potential experimental verification.

Contribution

It presents the geometric momentum as a new quantum operator for particles on curved surfaces, extending the geometric potential concept and aligning with Dirac's quantization.

Findings

Analytical expressions for geometric momentum distributions on spherical harmonics.

Compatibility of geometric momentum with Dirac's canonical quantization.

Potential experimental tests using rotational states of spherical molecules like C60.

Abstract

When a two-dimensional curved surface is conceived as a limiting case of a curved shell of equal thickness d, where the limit d\rightarrow0 is then taken, the well-known geometric potential is induced by the kinetic energy operator, in fact by the second order partial derivatives. Applying this confining procedure to the momentum operator, in fact to the first order partial derivatives, we find the so-called geometric momentum instead. This momentum is compatible with the Dirac's canonical quantization theory on system with second-class constraints. The distribution amplitudes of the geometric momentum on the spherical harmonics are analytically determined, and they are experimentally testable for rotational states of spherical molecules such as C_{60}.