Geometric Momentum and a Probe of Embedding Effects

As a manifold is embedded into a higher dimensional Euclidean space, quantum mechanics gives various embedding quantities. In the present study, two embedding quantities for a two-dimensional curved surface in the three-dimensional flat space, the geometric momentum and the geometric potential, are derived in a unified manner. For a particle moving on a two-dimensional sphere or a free rotation of a spherical top, the projections of both 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), thus constituting a dynamical (p(i), L-i) representation for the states on the two-dimensional spherical surface. The geometric momentum distribution of the states represented by spherical harmonics is successfully obtained, and this distribution for a homonuclear diatomic molecule seems within the resolution power of present momentum spectrometers and can be measured to probe the embedding effect.