Special rectangular (double-well and hole) potentials

TL;DR

This paper investigates special energy eigenstates in rectangular double-well and hole potentials, revealing previously overlooked zero-energy solutions that occur at specific potential depths or heights, expanding understanding of quantum bound states.

Contribution

It identifies new zero-energy eigenstates in rectangular potentials that occur at discrete potential parameters, which were previously missed in standard analyses.

Findings

Zero-energy states exist at specific potential parameters.

These states are described by linear wavefunctions of the form ψ(x)=Bx+C.

The results extend the known spectrum of rectangular potential problems.

Abstract

We revisit a rectangular barrier as well as a rectangular well (pit) between two rigid walls. The former is the well known double-well potential and the latter is a hole potential. Let $|V_0|$ be the height (depth) of the barrier (well) then for a fixed geometry of the potential, we show that in the double-well, $E=V_0(>0)$, and in the hole potential ($V_0 <0$), $E=0$, can be energy eigenvalues provided $V_0$ admits some special discrete values. These states have been missed out earlier which emerge only when one seeks the special zero-energy solution of one-dimensional Schr{\"o}dinger equation as $\psi(x)=Bx+C$.