Quaternionic bound states

TL;DR

This paper investigates bound states in quaternionic quantum mechanics, revealing that pure quaternionic potentials still support bound states and comparing these solutions to traditional complex potentials, advancing understanding of quaternionic effects.

Contribution

It demonstrates that pure quaternionic potentials do not eliminate bound states and provides initial analysis of quaternionic potentials' role in quantum phenomena.

Findings

Pure quaternionic potentials support bound states.

Bound state energies are discrete despite quaternionic nature.

Comparison with complex potentials highlights differences in bound state properties.

Abstract

We study the bound-state solutions of vanishing angular momentum in a quaternionic spherical square-well potential of finite depth. As in the standard quantum mechanics, such solutions occur for discrete values of energies. At first glance, it seems that the continuity conditions impose a very restrictive constraint on the energy eigenvalues and, consequently, no bound states was expected for energy values below the pure quaternionic potential. Nevertheless, a careful analysis shows that pure quaternionic potentials do not remove bound states. It is also interesting to compare these new solutions with the bound state solutions of the trial-complex potential. The study presented in this paper represents a preliminary step towards a full understanding of the role that quaternionic potentials could play in quantum mechanics. Of particular interest for the authors it is the analysis of confined wave packets and tunnelling times in this new formulation of quantum theory.