Constructing Hermitian Hamiltonians for spin zero neutral and charged particles on a curved surface: physical approach

Abstract

The Hamiltonians for a spin zero neutral particle and a charged particle in an electromagnetic field pinned to an arbitrary 2D surface embedded in 3D space are constructed. The new approach we follow starts with an expression for the 3D momentum operators whose components along the surface and the normal to the surface are separately Hermitian. The normal part of the kinetic energy operator is a Hermitian operator in this case. When this operator is dropped and the thickness of the layer is set to zero, one automatically gets the Hermitian surface Hamiltonian that contains the geometric potential term as expected. We show that Hermitian surface and normal momenta emerge automatically once one symmetrizes the usual normal and surface momentum operators; the geometrical potential originates from this symmetrization, too. Comparing our approach with the standard thin-layer quantization (TLQ), we show that Hermicity is also at the heart of TLQ.