A novel algebraic approach for the Schrödinger equation in split quaternionic mechanics

Abstract

In the theoretical exploration and numerical computation of split quaternionic quantum mechanics, an important goal is to solve the Schrödinger equation ∂∂t|f〉=−A|f〉 with A an i-Hermitian split quaternion matrix, and |f〉 an eigenstate to A. The split quaternionic Schrödinger equation is a fundamental equation in split quaternionic mechanics and plays an indispensable role. It is known that the problems of split quaternion Schrödinger equation can be discretized into the related eigen-problems of split quaternion matrix Aα=αλ. This paper studies the problem of split quaternion Schrödinger equation based on an isomorphic mapping, derives a novel approach for Schrödinger equation in split quaternion mechanics. This paper also shows an algebraic approach to the eigen-problems for i-Hermitian split quaternion matrices. The algebraic approach will have potential applications in the research field of split quaternionic mechanics.