Quantum Mechanics on Manifolds Embedded in Euclidean Space

Details Quantum Mechanics on Manifolds Embedded in Euclidean Space Quantum Mechanics on Manifolds Embedded in Euclidean Space

2003-02-27

arxiv.org/abs/hep-th/0302216v1

Annals Phys. 307 (2003) 132-143

Physics

High Energy Physics

High Energy Physics - Theory

12 pages, LaTeX

Scientific paper

10.1016/S0003-4916(03)00080-0

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.

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