Infinite Square Well, Self-Adjointness, and the Dirac Delta Function

One hundred years ago, Dirac presented a groundbreaking method for quantizing classical Hamiltonians. Simplicity is one of the virtues of the procedure, but certain classes of problems require special care. Here, we discuss one such class, the quantization of confined systems, with emphasis on the one-dimensional square well. We examine two presentations of the problem. The first presentation restricts the analysis to the interval $$0\le x\le L$$ . The second presentation considers the entire x axis, but an infinite potential traps the particle in the region $$0\le x\le L$$ . In contrast with the typical treatment in quantum mechanics textbooks, the first presentation imposes nontrivial boundary conditions. Proper treatment of the problem in the second presentation demands the use of delta functions or distributions.