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American Journal of Physics

An algebraic method for solving central force problems

Teman H. Cooke, J.L. Wood

Published Aug 13, 2002• 18 citations
Publisher

Abstract

A simple algebraic method, which is as easy to use as the angular momentum algebra, is demonstrated as a pedagogical way to solve certain central force problems exactly. Solutions for the hydrogen atom and the three-dimensional isotropic harmonic oscillator are presented together with a discussion of the limits of the method.
DOI
10.1119/1.1491262

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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.
Since the main purpose of the book was to demonstrate the philosophy and spirit behind the standard formulation of quantum mechanics, a thorough discussion of applications have taken a somewhat backseat. This was also done to keep the book accessible to a wide audience. Even outsiders interested to get a glimpse of quantum mechanics could get by with some minimal linear algebra provided in the book itself. Thus, technical and calculational details (which cannot be avoided if one were to discuss applications) had to be mostly avoided in the main text. However, in order that the book could still serve as an introductory text for a mainstream course, we make an exception in this chapter. Here we treat three of the mostwell known applications of quantum mechanics: Harmonic Oscillator, Angular Momentum and Hydrogen Atom. Both the analytical and the algebraic, ladder operator methods are demonstrated for solving the eigenvalue problems for the Harmonic oscillator and angular momentum. Topics involving technical calculations (e.g., calculations involving Hermite polynomials, spherical harmonics, etc.), modelling of real life systems (e.g., atomic polarizability) and spin have been relegated to the problem sets with elaborate hints and guidelines for working through them.