[Paper Review] Quantum particle constrained to a curved surface in the presence of a vector potential

M. Encinosa, Ray N. O'Neal|arXiv (Cornell University)|Aug 30, 1999

Quantum Mechanics and Non-Hermitian Physics2 references3 citations

TL;DR

This paper derives the Schrödinger equation for a charged quantum particle confined to a curved surface under a vector potential using differential forms. A key result is the emergence of a geometric coupling term between the normal component of the vector potential and the surface's mean curvature, which arises in the limit of infinitesimal confinement, revealing a non-trivial curvature-dependent interaction in constrained quantum systems.

ABSTRACT

The Schrodinger equation for a charged particle constrained to a curved surface in the presence of a vector potential is derived using the method of forms. In the limit that the particle is brought infinitesimally close to the surface, a term arises that couples the component of the vector potential normal to the surface to the mean curvature of the surface.

Motivation & Objective

• To derive the effective quantum Hamiltonian for a charged particle constrained to a curved surface in the presence of a vector potential.
• To investigate how geometric curvature influences the quantum dynamics of a confined particle under gauge fields.
• To determine the role of the vector potential's normal component to the surface in the low-confinement limit.
• To establish a rigorous field-theoretic framework using differential forms for constrained quantum systems.
• To identify novel geometric couplings arising from the interplay between curvature and gauge fields.

Proposed method

• The method of differential forms is employed to systematically derive the effective Schrödinger equation on the surface.
• The particle is considered to be confined to the surface in the limit of infinitesimal distance from it, using a constraint formalism.
• The vector potential is decomposed into tangential and normal components relative to the surface.
• The geometric structure of the surface, particularly its mean curvature, is incorporated via the Levi-Civita connection and metric tensor.
• The derivation yields a modified kinetic term that includes a coupling between the normal component of the vector potential and the surface's mean curvature.
• The resulting Hamiltonian includes a curvature-dependent term that modifies the effective dynamics of the particle.

Experimental results

Research questions

• RQ1 How does the presence of a vector potential affect the quantum dynamics of a particle constrained to a curved surface?
• RQ2 What geometric terms emerge in the effective Hamiltonian when the particle is confined to a surface with non-zero curvature?
• RQ3 How does the normal component of the vector potential couple to the surface's curvature in the limit of infinitesimal confinement?
• RQ4 What is the role of differential forms in deriving the constrained quantum dynamics on curved manifolds?
• RQ5 Can a curvature-dependent interaction be derived that couples the vector potential to the surface's mean curvature?

Key findings

• A novel coupling term emerges between the normal component of the vector potential and the mean curvature of the surface in the effective Hamiltonian.
• This curvature-gauge coupling arises naturally in the limit of infinitesimal confinement, indicating a geometrically induced interaction.
• The derived Schrödinger equation is consistent with the method of differential forms and respects the intrinsic geometry of the surface.
• The normal component of the vector potential contributes to the effective potential energy through the mean curvature, modifying the particle's dynamics.
• The result generalizes the standard constrained quantum mechanics by including gauge field effects in a geometrically consistent manner.
• The formalism provides a foundation for studying quantum Hall effects and anyonic statistics on curved surfaces with gauge fields.