A Closed Formula for the Barrier Transmission Coefficient in Quaternionic Quantum mechanics

TL;DR

This paper derives a closed-form expression for the transmission coefficient of a particle encountering a quaternionic potential barrier, enabling analytical insights into quaternionic quantum mechanics that were previously limited to numerical studies.

Contribution

It introduces a matrix-based method to obtain an explicit formula for the transmission coefficient in quaternionic quantum mechanics, advancing analytical understanding of such systems.

Findings

Provides a closed formula for the quaternionic transmission coefficient

Enables analysis of qualitative differences between complex and quaternionic quantum mechanics

Facilitates discussion of transmission times using stationary phase method

Abstract

In this paper, we analyze, by using a matrix approach, the dynamics of a non-relativistic particle in presence of a quaternionic potential barrier. The matrix method used to solve the quaternionic Schrodinger equation allows to obtain a closed formula for the transmission coefficient. Up to now, in quaternionic quantum mechanics, almost every discussion on the dynamics of non-relativistic particle was motived by or evolved from numerical studies. A closed formula for the transmission coefficient stimulates an analysis of qualitative differences between complex and quaternionic quantum mechanics, and, by using the stationary phase method, gives the possibility to discuss transmission times.