Self-adjointness and spontaneously broken symmetry

The notion of self-adjointness as distinct from Hermiticity is usually avoided in elementary quantum mechanics courses due to a feeling that it has nothing to do with physics or else is too mathematical. This is definitely not the case, and we use the model of a free particle restricted to motion on a line over a finite interval, to demonstrate this. For such a particle, there is one-parameter infinity of self-adjoint extensions of the momentum operator. We use one of these, corresponding to ’’antiperiodic’’ boundary conditions, to show that physical phenomena different from the usual case (periodic boundary conditions) occur. In fact, in this case, we have a model displaying many of the properties of spontaneously broken symmetry.