A note on the quantum Hamiltonian for a free particle

Benn, Ian and Tucker, Robin (1991) A note on the quantum Hamiltonian for a free particle. Physics Letters A, 156 (1-2). pp. 1-4. ISSN 0375-9601

Abstract

We consider the non-relativistic quantum mechanics of a free particle on a Riemannian manifold and offer an argument for an invariant quantum Hamiltonian proportional to the Laplacian amended by one quarter of the scalar curvature of the manifold. An explicit ordering of the arbitrary canonical variables in the classical Hamiltonian is found that implies the existence of a quantum potential that can be made to vanish at any point by adopting geodesic coordinates for that region of the manifold.