Momentum spectrometry of spherical harmonics and a probe of geometric embedding effect

TL;DR

This paper explores the measurement of geometric momentum distributions in spherical systems, proposing that current spectrometry techniques can detect embedding effects predicted by quantum mechanics for particles on curved surfaces.

Contribution

It introduces a (p_{i},L_{i}) representation for states on spherical surfaces and suggests feasible experimental probing of geometric embedding effects using existing momentum spectrometers.

Findings

Geometric momentum and angular momentum form a complete commuting set.

Ground state momentum distribution can be measured with current spectrometers.

Embedding effects influence the quantum states on curved surfaces.

Abstract

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.