Exploring properties of quantum eigenvalue sequences in an infinite square well with delta functions

Recent work carried out by undergraduate students at Berry College has shown that a simple quantum system consisting of an infinite square well (ISW) with Dirac delta functions placed inside can illustrate a rich variety of quantum phenomena. An ISW with regularly placed delta wells can have states of positive, negative, or even zero energy. The condition for the existence of zero-energy states provides a useful tool for exploring changes in the number of negative-energy states that are bound to the delta wells. An ISW with N regularly placed delta barriers has a pattern of energy level spacings that repeats every N+1 spacings. This result can be used to illustrate the formation of bands and gaps in the energy spectrum, as well as how the spectrum is altered by a defect in which one delta barrier has a different strength from the rest. If the barriers are placed irregularly, then there is no repeating pattern but the level spacings can be analyzed statistically, with results that depend on the probability T that a particle tunnels through one of the delta barriers. Even with a relatively small number of barriers we find that the spacings distribution nearly matches the Poisson distribution for T near 0, the Gaussian distribution for T close to 1, and the Gaussian Orthogonal Ensemble (GOE) distribution for some intermediate value of T. Starting with regularly placed barriers and gradually introducing disorder illustrates the transition from a repeating pattern of spacings to a non-repeating sequence that follows one of the standard distributions (Poisson, GOE, or Gaussian). We plan to continue our exploration of this simple model to illustrate even more interesting quantum phenomena.