Robin boundary conditions are generic in quantum mechanics

TL;DR

This paper demonstrates that Robin boundary conditions are more natural and broadly applicable in quantum mechanics than the traditionally used Dirichlet conditions, especially in modeling short-range potentials.

Contribution

It shows that Robin boundary conditions are useful in the long-wavelength approximation of short-ranged potentials, challenging the common preference for Dirichlet conditions.

Findings

Robin boundary conditions are effective in modeling short-range potentials.

Derived scattering and bound states for multi-step and Morse potentials.

Method applicable to various real quantum systems.

Abstract

Robin (or mixed) boundary conditions in quantum mechanics have received considerable attention in the last two decades, in particular, for applications to nanoscale systems. However, their utility has remained obscure to the larger physics community; in fact, one may find extant claims in the literature about the supposed naturalness of the specific Dirichlet (or vanishing) boundary condition. Here we demonstrate that not only are Dirichlet boundary conditions unnatural, but that Robin boundary conditions have utility in the long-wavelength approximation of short-ranged potentials. For illustration, we consider a non-relativistic particle in one dimension under the influence of the multi-step and Morse potential. We derive the scattering and bound states for these potentials and determine the parameters that describe those states in an effective system, namely one in which there is a free particle with a boundary. This method appears to be generically applicable and also generalizable to many other real systems.