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Effects of walls

We analyze here the energy states and associated wave functions available to a particle acted upon by a delta function potential of arbitrary strength and sign and fixed anywhere within a one-dimensional infinite well. We consider how the allowed energies vary with the well's width and with the location of the delta function within it. The model subtly distinguishes between whether the delta function is located at rational or irrational fractions of the well's width: in the former case all possible energy eigenvalues are solutions to a straightforward dispersion relation, but in the latter case, to make up a complete set these `ordinary' solutions must be augmented by the addition of `nodal' states which vanish at the delta function and so do not `see' it. Thus, although the model is a simple one, due to its singular nature it needs a little careful analysis. The model, of course, can be thought of as a limit of more physical smooth potentials which, though readily succumbing to straightforward numerical computation, would give little generic information. PACS numbers: 03.65.-w, 73.21.Fg, 01.40.-d