ABSTRACT
In the past few chapters, we have developed the Schrödinger equation and applied it to the solution of a number of quantum mechanical problems, mainly to develop experience with the results of simple and common systems. The results from these examples can be summarized relatively simply. In general, quantization enters the problem through the noncommuting nature of conjugate operators such as position and momentum. This has led to the Schrödinger equation itself as the primary equation of motion for the wave function solution to the problem. In essence, the system (e.g., the electron) is treated as a wave, rather than as a particle, and the wave equation of interest is the Schrödinger equation. When boundary conditions are applied, either through potential barriers, or through the form of the potential (as in the harmonic oscillator), the time-independent Schrödinger equation yields solutions that are often special functions. Examples of this are the sines and cosines in the rectangular-barrier case, the Airy functions in the triangular-well case, and the Hermite polynomials in the harmonic oscillator case.
