On Robin boundary conditions and the Morse potential in quantum mechanics

TL;DR

This paper investigates the origin of Robin boundary conditions in quantum mechanics, showing how they emerge from Morse potentials and phase-space formulations, and clarifies their relation to standard quantum walls.

Contribution

It demonstrates how Robin boundary conditions arise from Morse potentials in both Schrödinger and phase-space quantum mechanics, and clarifies their physical origin and relation to standard walls.

Findings

Robin boundary conditions can emerge from Morse potentials.

Dirichlet conditions require mass-dependent fine tuning.

Wigner functions reduce correctly in the contact interaction limit.

Abstract

The physical origin is investigated of Robin boundary conditions for wave functions at an infinite reflecting wall. We consider both Schr\"odinger and phase-space quantum mechanics (a.k.a. deformation quantization), for this simple example of a contact interaction. A non-relativistic particle moving freely on the half-line is treated as moving on the full line in the presence of an infinite potential wall, realized as a limit of a Morse potential. We show that the wave functions for the Morse states can become those for a free particle on the half-line with Robin boundary conditions. However, Dirichlet boundary conditions (standard walls) are obtained unless a mass-dependent fine tuning (to a reflection resonance) is imposed. This phenomenon was already observed for piece-wise flat potentials, so it is not removed by smoothing. We argue that it explains why standard quantum walls are standard. Next we consider the Wigner functions (the symbols of both diagonal and off-diagonal density operator elements) of phase-space quantum mechanics. Taking the (fine-tuned) limit, we show that our Wigner functions do reduce to the expected ones on the half-line. This confirms that the Wigner transform should indeed be unmodified for this contact interaction.