  Schroedinger developed an operator method for solving quantum mechanics.
While this technique is overshadowed by his more familiar differential equation
approach, it has found wide application as an illustration of supersymmetric
quantum mechanics. One reason for the reticence in its usage for conventional
quantum instruction is that the approach for simple problems like the
particle-in-a-box is much more complicated than the differential equation
approach, making it appear to be less useful for pedagogy. We argue that the
approach 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 show
how Schroedinger's operator method can be streamlined for these
particle-in-a-box problems greatly reducing the complexity of the solution and
making it much more accessible. As an application, we illustrate how this
approach can be used to prove an important result, the existence of bound
states for one- and two-dimensional attractive potentials, using only algebraic
methods. The approach developed here can be employed in undergraduate classes
and possibly even high school classes because it employs only algebra and
requires essentially no calculus.
