Solution of quantum eigenvalue problems by means of algebraic consistency conditions

TL;DR

This paper introduces an algebraic method for solving quantum eigenvalue problems that avoids solving the Schrödinger equation, using consistency conditions on operators to determine spectra.

Contribution

It presents a simple, pedagogical algebraic approach based on consistency conditions, applicable without eigenfunctions, for solving quantum eigenvalue problems.

Findings

Successfully applied to textbook quantum problems

Determines spectra without solving differential equations

Suitable for educational purposes

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 Schr\"odinger 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.