A general framework is presented of quantizing the system of a simple harmonic oscillator. Various methods of quantization so far proposed are accommodated here in a unified form. Some remarks are also made on quantization for the case of more than one oscillator.
Now that we have learned some of the principles which apply in quantum mechanics, we move on to the next more difficult problem of a quantized harmonic oscillator. The goal here is to provide a rigorous application of the polynomial method of solving differential equations on a relatively simple case and to provide some insight into how the Schrödinger equation was first solved [1]. Then we proceed to application in the form of worked examples. We have tried to give sufficient details of this solution to allow a student to follow the derivation with pencil and paper but do not forget to ponder over the spectroscopic applications! This author would agree that it is more important to absorb the main conclusions of this material than to master the derivations. In fact, the highest recommendation of this author is to always ask ‘‘What does this mean?’’ and absorb the conclusions for a future activity called ‘‘thinking’’ rather than just memorizing facts.
Two methods for obtaining the ladder spectrum characteristic of the simple harmonic oscillator are widely used in elementary quantum mechanics. In the first method, that of series solution, quantization is shown to follow from the requirement that the wave function be normalizable. In the second, it is common to derive quantisation in an operator formalism based on factorization of the wave equation. In this latter method the role of normalizability is not customarily discussed. We show explicitly that the same boundary condition implicit in the operator manipulations leads to quantization in both methods.