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Cambridge University Press eBooks

The Harmonic Oscillator

Rocco Schiavilla

Published Dec 5, 2024• 0 citations
Publisher

Abstract

The lowering, raising, and number operators are defined and their properties are studied; the eigenfunctions of the harmonic oscillator Hamiltonian are derived; coherent or quasi-classical states are obtained and their properties examined. Several applications are discussed, including, among others, the isotropic harmonic oscillator in N dimensions and a model for a one-dimensional crystal.

DOI
10.1017/9781009473637.009

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The lowering, raising, and number operators are defined and their properties are studied; the eigenfunctions of the harmonic oscillator Hamiltonian are derived; coherent or quasi-classical states are obtained and their properties examined. Several applications are discussed, including, among others, the isotropic harmonic oscillator in N dimensions and a model for a one-dimensional crystal.
Chapter 9 is devoted to a detailed study of the harmonic oscillator in one dimension. The time-dependent Schrödinger equation is used to construct the coherent oscillator packet with minimal dispersion, which provides an opportunity to appreciate the similarities and differences between the dynamics of the classical and quantum oscillators. In a second step, we study the stationary solutions of the oscillator, which represent one of the most important orthogonal sets of quantum eigenfunctions because of its many applications, properties, and relative simplicity. Further, the use of raising and lowering (or creation and annihilation) operators is shown to be a powerful way of dealing with both material and field oscillators in general. Finally, the problem of two coupled oscillators is used to generalize the Schrödinger equation to the case of more than one particle.
Abstract In both classical and quantum physics the harmonic oscillator is one of the most important physical systems [1–4]. Furthermore, because of the simplicity and symmetry of the Hamiltonian operator, the quantum mechanical oscillator is particularly suited to analysis by abstract operator methods. For this reason the study of the one dimensional oscillator is a useful introduction to the application of operator methods in quantum mechanics. In Sections 2.1–2.4 we use these methods to determine energy eigenvalues and eigenkets, and matrix elements; the analysis is extended to isotropic multidimensional oscillators in Section 2.10. Quasiclassical states of the oscillator are discussed in Section 2.5. In Section 2.6 shift operators are used to calculate wave functions. In Sections 2.7–2.11 we discuss some applications of the theory of the oscillator.
The isotropic harmonic oscillator in N dimensions is shown to have an underlying symmetry group O(2,1)X O(N)which implies a unique result for the energy spectrum of the system. Raising and lowering operators analogous to those of the one-dimensional oscillator are given for each value of the angular momentum parameter. This allows the construction of an infinite number of coherent states to be carried out. In the N=1 case there is a twofold family of coherent states, a particular linear combination of which coincides with the single set already well known for that case. Wave functions are readily derived which require only the solution of a first order differential equation, an attribute generally characteristic of group theoretical approaches.
The isotropic harmonic oscillator in N dimensions is shown to have an underlying symmetry group O(2,1)×O(N ) which implies a unique result for the energy spectrum of the system.Raising and lowering operators analogous to those of the one-dimensional oscillator are given for each value of the angular momentum parameter.This allows the construction of an infinite number of coherent states to be carried out.In the N = 1 case there is a twofold family of coherent states, a particular linear combination of which coincides with the single set already well known for that case.Wave functions are readily derived which require only the solution of a first order differential equation, an attribute generally characteristic of group theoretical approaches.