An exactly solvable dead-layer problem

TL;DR

This paper provides an exact solution to a one-dimensional two-particle system with delta-function attraction, revealing that the center-of-mass moves in a reduced volume due to a dead layer near the walls, with the dead-layer width related to the molecule size.

Contribution

It introduces an exact solution to the dead-layer problem in a one-dimensional quantum system and characterizes the dead-layer width's dependence on system parameters.

Findings

Dead-layer width equals the molecule size for large volumes.

Dead-layer width is smaller than the molecule size for finite volumes.

Dead-layer width appears independent of particle mass differences.

Abstract

A molecule consisting of two particles interacting with a delta-function attraction, and confined to a one-dimensional volume, is studied. From the exact solution of the system we deduce that the center-of-mass effectively moves in a volume reduced by an inaccessible dead layer near each wall. For a very large volume the dead-layer width equals the size of the molecule, defined as the expectation value of the interparticle distance for the unconfined molecule. For finite volumes the dead-layer width is smaller than this. For unequal particle masses, perturbation in the mass difference up to third order yields the same result, and we conjecture that for this system the dead-layer width is independent of the mass values.