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arXiv:2110.03829 · European Journal of Physics

Solution of quantum eigenvalue problems by means of algebraic consistency conditions

Luis de la Peña, Ana Marı́a Cetto, Andrea Valdés-Hernández

Published Oct 11, 2021• 0 citations
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Abstract

Abstract We present a simple algebraic procedure that can be applied to solve a range of quantum eigenvalue problems without the need to know the solution of the Schrödinger equation. The procedure, presented with a pedagogical purpose, is based on algebraic consistency conditions that must be satisfied by the eigenvalues of a couple of operators proper of the problem. These operators can be either bilinear forms of the raising and lowering operators appropriate to the problem, or else auxiliary operators constructed by resorting to the factorization of the Hamiltonian. Different examples of important quantum-mechanical textbook problems are worked out to exhibit the clarity and simplicity of the algebraic procedure for determining the spectrum of eigenvalues without knowing the eigenfunctions. For this reason the material presented may be particularly useful for undergraduate students or young physicists.

DOI
10.1088/1361-6404/ac2ecd

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Abstract We present a simple algebraic procedure that can be applied to solve a range of quantum eigenvalue problems without the need to know the solution of the Schrödinger equation. The procedure, presented with a pedagogical purpose, is based on algebraic consistency conditions that must be satisfied by the eigenvalues of a couple of operators proper of the problem. These operators can be either bilinear forms of the raising and lowering operators appropriate to the problem, or else auxiliary operators constructed by resorting to the factorization of the Hamiltonian. Different examples of important quantum-mechanical textbook problems are worked out to exhibit the clarity and simplicity of the algebraic procedure for determining the spectrum of eigenvalues without knowing the eigenfunctions. For this reason the material presented may be particularly useful for undergraduate students or young physicists.
We show that it is possible to give to any (linear and Hermitian) operator with discrete spectrum, acting on a Hilbert space, a canonical form in terms of the commutator and anticommutator of a pair of adequately constructed raising and lowering operators. This construction enables one to investigate in a very compact way the (discrete) spectrum of the given operator, which becomes determined by a set of consistency conditions. We apply the method to several important problems of quantum mechanics.
A new method for treating the most important eigenvalue problems in quantum mechanics is developed. The solutions can be found immediately once the equations are factorized by means of linear operators. These operators acting on a normalized eigenfunction change it into a new normalized eigenfunction and all solutions can be found once the basic eigenfunction is known. This basic eigenfunction is a solution of a simple differential equation of the first order. The underlying theory is explained more fully on a special case (Section 1) and then the rules of procedure are formulated explicitly (Section 2). The rest of the paper contains applications including the Kepler problem treated according to Dirac's theory.
Abstract Quantum mechanical problems capable of exact solution are traditionally solved in a few instances only (such as the harmonic oscillator and angular momentum) by operator methods, but mainly by means of Schrodinger's wave mechanics. The present volume shows that a large range of one- and three-dimensional problems, including certain relativistic ones, are solvable by algebraic, representation-independent methods using commutation relations, shift operators, the virial, hypervirial, and Hellman-Feyman theorems. Applications of these operator methods to the calculation of eigenvalues, matrix elements, and wavefunctions are discussed in detail. This book is an introduction to the use of operator methods in quantum mechanics and also a reference work with numerous problmes solved. It is suitable for use by students of intermediate quantum mechanics and also more advanced postgraduate students who wish to study the algebraic method of solving quantum mechanical problems.
The harmonic oscillator is one of the most studied systems in Physics with a myriad of applications. One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator with analytic and algebraic approaches. In the algebraic solution, creation and annihilation operators are introduced to factorize the Hamiltonian. This work presents an algebraic solution for the simple harmonic oscillator in the context of Classical Mechanics, exploring the Hamiltonian formalism. In this solution, similarities between the canonical coordinates in a convenient basis for the classical problem and the corresponding operators in Quantum Mechanics are highlighted. Moreover, the presented algebraic solution provides a straightforward procedure for the quantization of the classical harmonic oscillator, motivating and justifying some operator definitions commonly used to solve the correspondent problem in Quantum Mechanics.
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Courses on undergraduate quantum mechanics usually focus on solutions of the Schrödinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced, only academic examples are used, such as the matrix representation of Dirac’s raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.