Analytic Plane Wave Solutions for the Quaternionic Potential Step

TL;DR

This paper derives explicit analytic solutions for quaternionic quantum systems with step potentials, highlighting differences from standard quantum mechanics and providing tools for future analytical and experimental investigations.

Contribution

It introduces a new analytic solution approach for quaternionic quantum systems, extending the mathematical toolkit beyond numerical methods.

Findings

Explicit quaternionic step potential solutions derived

Differences between quaternionic and complex quantum systems demonstrated

Potential implications for experimental tests of quantum mechanics

Abstract

By using the recent mathematical tools developed in quaternionic differential operator theory, we solve the Schroedinger equation in presence of a quaternionic step potential. The analytic solution for the stationary states allows to explicitly show the qualitative and quantitative differences between this quaternionic quantum dynamical system and its complex counterpart. A brief discussion on reflected and transmitted times, performed by using the stationary phase method, and its implication on the experimental evidence for deviations of standard quantum mechanics is also presented. The analytic solution given in this paper represents a fundamental mathematical tool to find an analytic approximation to the quaternionic barrier problem (up to now solved by numerical method).