Effects of walls

TL;DR

This paper investigates the quantum energy states of a particle in a one-dimensional infinite well with a delta function potential at arbitrary positions and strengths, revealing subtle distinctions based on the delta's location.

Contribution

It provides a detailed analysis of how a delta potential affects energy eigenvalues and wave functions, including the distinction between rational and irrational delta positions within the well.

Findings

Energy eigenvalues depend on the delta potential's position and strength.

Rational delta positions lead to straightforward dispersion relations.

Irrational positions require additional nodal states for completeness.

Abstract

We analyze here the energy states and associated wave functions available to a particle acted upon by a delta function potential of arbitrary strength and sign and fixed anywhere within a one-dimensional infinite well. We consider how the allowed energies vary with the well's width and with the location of the delta function within it. The model subtly distinguishes between whether the delta function is located at rational or irrational fractions of the well's width: in the former case all possible energy eigenvalues are solutions to a straightforward dispersion relation, but in the latter case, to make up a complete set these `ordinary' solutions must be augmented by the addition of `nodal' states which vanish at the delta function and so do not `see' it. Thus, although the model is a simple one, due to its singular nature it needs a little careful analysis. The model, of course, can be thought of as a limit of more physical smooth potentials which, though readily succumbing to straightforward numerical computation, would give little generic information. PACS numbers: 03.65.-w, 73.21.Fg, 01.40.-d