Avoided crossings of energy levels in one-dimension

TL;DR

This paper investigates the phenomenon of avoided crossings in energy levels of one-dimensional quantum systems, demonstrating that simple double-well potentials can exhibit this behavior when one well's width is varied.

Contribution

It shows that avoided crossings can occur in simple one-dimensional double-well potentials, expanding understanding beyond complex models.

Findings

Avoided crossings occur in simple double-well potentials.

Energy levels come close but do not cross as parameters vary.

Spectral properties depend on potential geometry.

Abstract

Due to the absence of degeneracy in one dimension, when a parameter, $\lambda$, of a potential is varied slowly the discrete energy eigenvalue curves, $E_n(\lambda)$, cannot cross but they are allowed to come quite close and diverge from each other. This phenomena is called avoided crossing of energy levels. The parametric evolution of eigenvalues of the generally known one dimensional potentials do not display avoided crossings and on the other hand some complicated and analytically unsolvable models do exhibit this. Here, we show that this interesting spectral property can be found in simple one-dimensional double-wells when width of one of them is varied slowly.