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Schroedinger developed an operator method for solving quantum mechanics.While this technique is overshadowed by his more familiar differential equationapproach, it has found wide application as an illustration of supersymmetricquantum mechanics. One reason for the reticence in its usage for conventionalquantum instruction is that the approach for simple problems like theparticle-in-a-box is much more complicated than the differential equationapproach, making it appear to be less useful for pedagogy. We argue that theapproach is still quite attractive because it employs only algebraic methods,and thereby has a much lower level of math background needed to use it. We showhow Schroedinger's operator method can be streamlined for theseparticle-in-a-box problems greatly reducing the complexity of the solution andmaking it much more accessible. As an application, we illustrate how thisapproach can be used to prove an important result, the existence of boundstates for one- and two-dimensional attractive potentials, using only algebraicmethods. The approach developed here can be employed in undergraduate classesand possibly even high school classes because it employs only algebra andrequires essentially no calculus.

Proving the existence of bound states for attractive potentials in 1-d and 2-d without calculus