Discussion

Is it worthwhile to note the existence of this ladder operator approach? That curious students might worry about this obvious omission (or absence) prompts these remarks. But there is no question that the presentation implies a circularity of reasoning which, although it might exist in other problems, is never so self-evident. We are presupposing the existence and form of the eigenfunctions prior to attacking the problem. Had we started with the operators as postulated in Equation 2, the circularity of the argument would not have been so evident.

What is worth keeping here? It is clear that the current repertoire of exercises of evaluating commutators can be expanded to include examples other than xⁿ and p_x^m e.g., $[p_x,\sin x]$.

Also, what becomes clear in constructing the original ladder operator, is that the trigonometric formula for the sin of the sum of two angles makes the form possible. This implies that similar ``addition'' formulae lie at the fundament of the ladder operator approach in more common cases, which is, perhaps, enlightening.

In all, however, this ladder operator approach to the particle in a box problems probably deserves its own obscurity (or absence). No simplification occurs in carrying out this derivation which warrants expending the effort to lay the ground work for the derivation.

One concludes that the ladder operator exists, is interesting, but that its relevance to the teaching and learning processes lies somewhere in the venue of exercises.