Quantum-Mechanical Harmonic Oscillator

Abstract

Quantum-mechanical treatment of a harmonic oscillator has been a well-studied topic from the beginning of the history of quantum mechanics. This topic is a standard subject in classical mechanics as well. In this chapter, first we briefly survey characteristics of a classical harmonic oscillator. From a quantum-mechanical point of view, we deal with features of a harmonic oscillator through matrix representation. We define creation and annihilation operators using position and momentum operators. A Hamiltonian of the oscillator is described in terms of the creation and annihilation operators. This enables us to easily determine energy eigenvalues of the oscillator. As a result, energy eigenvalues are found to be positive definite. Meanwhile, we express the Schrödinger equation by the coordinate representation. We compare the results with those of the matrix representation and show that the two representations are mathematically equivalent. Thus, the treatment of the quantum-mechanical harmonic oscillator supplies us with a firm ground for studying basic concepts of the quantum mechanics.