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European Journal of Physics

Computing quantum eigenvalues made easy

H. J. Korsch, Markus Glück

Published Jul 1, 2002• 33 citations
Publisher

Abstract

An extremely simple and convenient method is presented for computing eigenvalues in quantum mechanics by representing position and momentum operators in matrix form. The simplicity and success of the method is illustrated by numerical results concerning eigenvalues of bound systems and resonances for Hermitian and non-Hermitian Hamiltonians as well as driven quantum systems. Various MATLAB program codes are listed.

DOI
10.1088/0143-0807/23/4/305

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An extremely simple and convenient method is presented for computing eigenvalues in quantum mechanics by representing position and momentum operators in matrix form. The simplicity and success of the method is illustrated by numerical results concerning eigenvalues of bound systems and resonances for Hermitian and non-Hermitian Hamiltonians as well as driven quantum systems. Various MATLAB program codes are listed.
Abstract Convenient and simple numerical techniques for performing quantum computations based on matrix representations of Hilbert space operators are presented and illustrated by various examples. The applications include the calculations of spectral and dynamical properties for one-dimensional and two-dimensional single-particle systems as well as bosonic many-particle and open quantum systems. Due to their technical simplicity these methods are well suited as a tool for teaching quantum mechanics to undergraduates and graduates. Explicit implementations of the presented numerical methods in Matlab are given.
Operators in quantum mechanics which represent observables are of a special type: They are linear and Hermitian. Such operators not only form computation tools, but also form the most effective language in terms of which the theory can be formulated. This chapter presents important properties of linear operators and their eigenfunctions. It discusses the features of various operators which are necessary for studying quantum mechanical systems. The chapter shows how rich and beautiful the mathematical structure of quantum mechanics is. There exists a class of eigenvalue problems, the so-called parity-time reversal (PT) -symmetric non-Hermitian Hamiltonian problems. They have arisen in recent years in a number of physical contexts such as localization transitions, vortex pinning and population biology. Such non-Hermitian Hamiltonians can be used to describe certain experimental phenomena. PT-symmetry has been noticed in certain optical systems, such as optical couplers, microwave billiard and large-scale temporal lattices.
Classical physics began to be superseded by quantum physics because it could not explain microscopic phenomena. Quantum physics employs mathematical approaches such as wave mechanics and matrix mechanics, one of which is the eigenvalue problem involving matrix operators. There are many types of matrix operators; however, not all are suitable for measuring physical quantities in quantum physics. Operators in quantum physics must be Hermitian to ensure that their eigenvalues are real. Eigenvalue problems for matrix operators can be solved either analytically or numerically. Therefore, the objective of this research is to analyze symmetric and antisymmetric matrix operators in the measurement of physical quantities through eigenvalue problems. The research method utilized is a theoretical computational approach. The results demonstrate that symmetric matrix operators are Hermitian and yield real eigenvalues. In contrast, antisymmetric matrix operators are non Hermitian, yielding imaginary or complex eigenvalues. The results of solving eigen problems using Python obtained relative errors between 0 and orders., while the absolute error remained below 10%, thus confirming the validity of the results. Based on the findings, symmetric matrix operators satisfy the requirements for use in physical problems through eigenvalue analysis. The eigenvalues of antisymmetric matrices cannot represent the measurement of physical quantities directly and serve only as rotation operators. However, antisymmetric matrices can be utilized in quantum physics for physical measurements if they are transformed into Hermitian matrices. Furthermore, solving eigenvalue problems with Python is effective due to its high accuracy and small error margins.
This chapter illustrates how one can obtain a matrix representation of a state vector, an operator, and an eigenvalue equation in a basis of orthonormal states. It helps to learn that matrix representations are very useful in calculations of the properties of operators, because algebra of matrices is very simple and completely developed. Readers familiar with the algebra of matrices agree that mathematical operations on matrices such as multiplication by a scalar, addition, subtraction, multiplication and diagonalization are simple and easy to perform. In fact, the application of the algebra of matrices to quantum physics results from the direct relations between operators and matrices that any mathematical relation that holds between operators also holds between the corresponding matrices. The eigenvalues and eigenvectors of the matrix representing a given operator  in the basis of orthonormal states are the same as the eigenvalues and eigenvectors of Â.
The spectral analysis of a non-Hermitian unbounded operator appearing in quantum physics is our main concern. The properties of such an operator are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operator and its adjoint can be diagonalized when expressed in terms of certain conveniently constructed operators. We show that their eigenfunctions constitute complete systems, but do not form Riesz bases. Attempts to overcome this difficulty in the quantum mechanical set up are pointed out.