Quantum Mechanics on Manifolds Embedded in Euclidean Space

TL;DR

This paper reviews the formalism of quantum particles confined to manifolds embedded in higher-dimensional Euclidean spaces, focusing on the effects of geometry-induced gauge potentials and the distinction between intrinsic and extrinsic geometry.

Contribution

It provides a comprehensive review of the confining potential formalism for quantum particles on arbitrary embedded manifolds, highlighting the role of gauge potentials and geometric effects.

Findings

Identification of gauge potentials only when normal state space is degenerate

Explicit expressions for scalar potentials induced by geometry

Analysis of a 3D manifold embedded in 5D Euclidean space

Abstract

Quantum particles confined to surfaces in higher dimensional spaces are acted upon by forces that exist only as a result of the surface geometry and the quantum mechanical nature of the system. The dynamics are particularly rich when confinement is implemented by forces that act normal to the surface. We review this confining potential formalism applied to the confinement of a particle to an arbitrary manifold embedded in a higher dimensional Euclidean space. We devote special attention to the geometrically induced gauge potential that appears in the effective Hamiltonian for motion on the surface. We emphasize that the gauge potential is only present when the space of states describing the degrees of freedom normal to the surface is degenerate. We also distinguish between the effects of the intrinsic and extrinsic geometry on the effective Hamiltonian and provide simple expressions for the induced scalar potential. We discuss examples including the case of a 3-dimensional manifold embedded in a 5-dimensional Euclidean space.