An algebraic method for Schrödinger equations in quaternionic quantum mechanics☆
Introduction
In recent year, there has been a wide interest in formulating quantum theories by using the non-commutative field of quaternions [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. In the study of theory and numerical computations of quaternionic quantum mechanics and field theory, one of the most important tasks is to solve the Schrödinger equation with A an anti-self-adjoint real quaternion matrix, and an eigenstate to A. The quaternionic Schrödinger equation (1.1) plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger equation (1.1) is reduced to the study of quaternionic eigen-equation with A an anti-self-adjoint real quaternion matrix (time-independent).
In the study of theory and numerical computations of quaternionic quantum theory, in order to well understand the perturbation theory, experimental proposals and theoretical discussions underlying the quaternionic formulations of the Schrödinger equation and so on, one often meets problems of approximate solutions of quaternion problems, such as approximate solution of quaternion linear equations that is appropriate when there are errors in the vector α and λ, i.e. quaternionic Least Squares eigenproblem (QLSE) in quaternionic quantum mechanics.
The main difficulty in obtaining the quaternionic approximate solutions of a physical problem is due to the fact of the non-commutation of quaternion in general, and the standard mathematical methods of resolution break down. It is known that the complex Least Squares eigenproblem (LSE) has been developed as a global fitting technique especially in physics for solving approximate solutions of complex linear equations if errors occur in the vector α and λ. But the QLSE problem has not been settled now. In this paper, by means of complex representation, we study the quaternionic Least Squares eigenproblem (QLSE), and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.
Let R denote the real number field, C the complex number field, Q the quaternion number field. For any quaternion in which . For any quaternion where , the conjugate of quaternion x is , and . For any quaternion matrix A, , and denote the transpose, conjugate and conjugate transpose of A over quaternion field, respectively. denotes the set of matrices on a field F. For any quaternion matrix , , , A can be uniquely written as , . It is easy to verify that for any , we have , and . For any , A is unitary if ; and A is anti-self-adjoint if . Let , a quaternion λ is said to be a right (left) eigenvalue provided that , and α is said to be an eigenvector to corresponding eigenvalue λ. Two quaternions x and y are said to be similar if there exists a nonzero quaternion p such that , and this is written as . It is routine to check that ∼ is an equivalence relation on the quaternions. We denote by the equivalence class containing x.
By [14, Proposition 2.1] we easily get following result.
Lemma 1.1 Let be a real quaternion. Then there exists a unit quaternion such that namely . The complex number in (1.5) is called principal number of the class .
Section snippets
Norms of quaternion matrices
In this section, we introduce concepts of norms of quaternion matrices by means of complex representation of quaternion matrices.
For any quaternion matrix , in paper [14], the author defined a complex representation the complex matrix was called complex representation of A.
Let , , . From [14] we have following results. in which is a unitary matrix, is identity
Quaternionic Least Squares eigenproblem
In this section, by means of complex representation, we study the quaternionic Least Squares eigenproblem (QLSE), and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.
In the study of theory and practical numerical computations in quaternionic quantum mechanics, the eigenproblem of finding eigenvalues and eigenvectors of a quaternion matrix is a very important and difficult problem. In practical,
Algorithm
In last section, Theorem 3.4 sets up a bridge between the solutions of the quaternionic Least Squares eigenproblem (3.1) and that of the complex Least Squares eigenproblem (3.5), and suggests an algebraic technique of computing a solution of quaternionic Least Squares eigenproblem (3.1) by that of complex Least Squares eigenproblem (3.5). In this section, we list an algorithm for computing a solution of quaternionic Least Squares eigenproblem (3.1) by means of complex representation.
Algorithm Let .
Conclusions
In this paper, by means of complex representation of a quaternion matrix, we first introduce the norms of quaternion matrices, study the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix. This paper sets up a bridge between the solutions of the quaternionic Least Squares eigenproblem and that of the complex Least Squares eigenproblem, turns the problems of solutions of
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This paper is partly supported by the National Natural Science Foundation of China (10671086) and Shandong Natural Science Foundation of China (Y2005A12).