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Journal of Chemical Education

On the genesis of ladder operators

A.B. Sannigrahi

Published Mar 1, 1985• 0 citations
Publisher

Abstract

Justifies the choice of ladder operators in simplifying the quantum mechanical treatment of angular momentum and linear harmonic oscillator problems.
DOI
10.1021/ed062p205

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The factorization of four vector operators, D±(ω) and D±(−ω), which occur in a representation-independent, spectrum-generating algebra for the three-dimensional, isotropic harmonic oscillator in an angular momentum basis, is considered (ω is the angular frequency of the oscillator). The D±(ω) are quantum-mechanical analogs of the classical vectors (1∓iL̂×)Fc (ω), where Fc(ω)=−Mωr×L+pL is constant in a frame rotating with angular velocity ωL̂. It is shown that these four vector operators can be factorized in two different ways to yield operators that, apart from their dependence on a constant of the motion (L2), are linear in either p or r. In this way 20 abstract operators are obtained. The properties of these operators are discussed: (i) Twelve are ladder operators for the quantum numbers l, and l and m, in the eigenkets ‖lm〉 of L2 and Lz. In linearized, differential form six of these operators are ladder operators for the spherical harmonics in the coordinate representation, while the other six are the corresponding operators in the momentum representation. (ii) The remaining eight operators factorize linear combinations of the Hamiltonian and the dimension operator. In linearized, differential form four of these operators are ladder operators for energy and angular momentum in the radial part of the coordinate-space wave functions, while the other four are the corresponding operators in the momentum representation.
Abstract Modern applications of quantum mechanics are illustrated, and the simplifications offered by operator and algebraic techniques are highlighted. The various "pictures" (Schrödinger, Heisenberg, and Interaction or Dirac) are discussed and their origins are motivated. It is shown that operator techniques simplify the harmonic oscillator and angular momentum through the use of ladder operators. The WKB approximation boundary conditions are developed through the use of complex number algebra. Other topics that benefit from these techniques are discussed, including the squeezing of optical fields and various alternative solutions to the hydrogen atom.
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.