An algebraic method for Schrödinger equations in quaternionic quantum mechanics

Abstract

In the study of theory and numerical computations of quaternionic quantum mechanics and quantum chemistry, one of the most important tasks is to solve the Schrödinger equation ∂ ∂ t | f 〉 = − A | f 〉 with A an anti-self-adjoint real quaternion matrix, and | f 〉 an eigenstate to A. The quaternionic Schrödinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger equation is reduced to the study of quaternionic eigen-equation A α = α λ with A an anti-self-adjoint real quaternion matrix (time-independent). This paper, by means of complex representation of quaternion matrices, introduces concepts of norms of quaternion matrices, studies the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.