The ground state energy of a charged particle on a Riemann surface

TL;DR

This paper derives a universal formula for the quantum ground state energy of a charged particle on any compact Riemann surface under a constant magnetic field, showing independence from geometric and topological details.

Contribution

It introduces a novel reinterpretation of the quantum Hamiltonian as a second variation operator, leading to a universal energy formula.

Findings

Ground state energy is E_0=eB/2m for any Riemann surface.

The energy formula is independent of surface geometry and topology.

The approach links quantum Hamiltonian to classical variational problems.

Abstract

It is shown that the quantum ground state energy of particle of mass m and electric charge e moving on a compact Riemann surface under the influence of a constant magnetic field of strength B is E_0=eB/2m. Remarkably, this formula is completely independent of both the geometry and topology of the Riemann surface. The formula is obtained by reinterpreting the quantum Hamiltonian as the second variation operator of an associated classical variational problem.