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Acta Physica Sinica

On the Factorization Method for Quantum Mechanical Eigenvalue Problems

H.C.Lee

Published Jan 1, 1944• 0 citations
Publisher

Abstract

The Schrodinger-Infold factorization method for salrinz eigenralue problem in quantum mechanics is very simple for the determination of the sigenvalues and oigenfunctione, without usinx power series or polguomial developments. Howerer, no general theory underlying thie method has been depuloned. It is the parpose of tho preseat paper to formulate explicit conditiona (A, R, G) ander which the method applies, and to lay down formulas which will enable one to write ont at once the solution of ans eigenvalue problem eatisfying these conditions. As illastrationa tho spherical harmonics, the Kepler problem in ordinary space and in spherical epaco, and the harmonie oscillator, are re-troated here on the basis of the theory now developed.

DOI
10.7498/aps.5.89

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The Schrodinger-Infold factorization method for salrinz eigenralue problem in quantum mechanics is very simple for the determination of the sigenvalues and oigenfunctione, without usinx power series or polguomial developments. Howerer, no general theory underlying thie method has been depuloned. It is the parpose of tho preseat paper to formulate explicit conditiona (A, R, G) ander which the method applies, and to lay down formulas which will enable one to write ont at once the solution of ans eigenvalue problem eatisfying these conditions. As illastrationa tho spherical harmonics, the Kepler problem in ordinary space and in spherical epaco, and the harmonie oscillator, are re-troated here on the basis of the theory now developed.
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.
The factorization method of Schrödinger shows us how to determine the energy eigenstates without needing to determine the wavefunctions in position or momentum space. A strategy to convert the energy eigenstates to wavefunctions is well known for the one-dimensional simple harmonic oscillator by employing the Rodrigues formula for the Hermite polynomials in position or momentum space. In this work, we illustrate how to generalize this approach in a representation-independent fashion to find the wavefunctions of other problems in quantum mechanics that can be solved by the factorization method. We examine three problems in detail: (i) the one-dimensional simple harmonic oscillator; (ii) the three-dimensional isotropic harmonic oscillator; and (iii) the three-dimensional Coulomb problem. This approach can be used in either undergraduate or graduate classes in quantum mechanics.
The solution of the quantum-mechanical eigenvalue problem is discussed for cases when a series of approximate eigenfunctions is known. If these ``unperturbed'' states are divided into two classes, a perturbation formula is derived giving the influence of one class of states on the other in the final solution. The formula contains as special cases: (i) the Schrödinger-Brillouin formula for the eigenvalue of a nondegenerate state, (ii) a new simple formula for treating a class of degenerate states, and (iii) the splitting of the secular equation in cases where the system naturally consists of two independent parts in mutual interaction.
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.
This original and innovative textbook takes the unique perspective of introducing and solving problems in quantum mechanics using linear algebra methods, to equip readers with a deeper and more practical understanding of this fundamental pillar of contemporary physics. Extensive motivation for the properties of quantum mechanics, Hilbert space, and the Schrödinger equation is provided through analysis of the derivative, while standard topics like the harmonic oscillator, rotations, and the hydrogen atom are covered from within the context of operator methods. Advanced topics forming the basis of modern physics research are also included, such as the density matrix, entropy, and measures of entanglement. Written for an undergraduate audience, this book offers a unique and mathematically self-contained treatment of this hugely important topic. Students are guided gently through the text by the author's engaging writing style, with an extensive glossary provided for reference and numerous homework problems to expand and develop key concepts. Online resources for instructors include a fully worked solutions manual and lecture slides.
Two methods for obtaining the ladder spectrum characteristic of the simple harmonic oscillator are widely used in elementary quantum mechanics. In the first method, that of series solution, quantization is shown to follow from the requirement that the wave function be normalizable. In the second, it is common to derive quantisation in an operator formalism based on factorization of the wave equation. In this latter method the role of normalizability is not customarily discussed. We show explicitly that the same boundary condition implicit in the operator manipulations leads to quantization in both methods.