ABSTRACT
One of the most commonly used quantum mechanical systems is that of the simple harmonic oscillator. The prototype system is the simple pendulum. Classically, the motion of the system exhibits harmonic, or sinusoidal oscillations. Quantum mechanically, the motion described by the Schrödinger equation is more complex. Although quite a simple system in principle, the harmonic oscillator assumes almost overwhelming importance due to the fact that it is one of the few systems that can be solved exactly in quantum mechanics. Almost any interaction potential may be expanded in a Tayor series with the low-order terms (up to quadratic order) cast into a form described by the harmonic oscillator. The sinusoidal motion of classical mechanics is the simplest system that produces oscillatory behavior, and therefore is found in almost an infinity of physical systems. Thus, the properties of the harmonic oscillator become important for their use in describing quite diverse physical systems. In this chapter, we will develop the general mathematical solution for the wave functions of the harmonic oscillator, using a simple operator algebra that allows us to obtain these wave functions and properties in a much more usable and simple manner. Classically, the harmonic oscillator is described in terms of Hermite polynomials, as the differential equation is one of the variants of the Sturm-Liouville problem. We can obtain these functions without the normal complication of solving the detailed differential equation, as the operator approach makes life easier. We then turn to two classic examples of systems in which we use the results for the harmonic oscillator to explain the properties: the simple LC-circuit and vibrations of atoms in a crystalline lattice. We the discuss motion in a quantizing magnetic field, as it is another system that reduces to a harmonic oscillator.
