Study of Particle Interactions in Quantum Systems

Quantum mechanical systems are characterized by their energy eigenvalues.~Previous~studies have shown that the distribution of spacings between adjacent energy eigenvalues is related to the dynamics in the classical version of the system.~Systems with regular dynamics have eigenvalue spacings~that follow the Poisson distribution, while systems with chaotic dynamics have spacings that~follow the Gaussian Orthogonal Ensemble~(GOE)~distribution.~ The goal of my research was to find a very simple quantum system that exhibits a transition from Poisson to GOE statistics, even though the classical dynamics doesn't clearly change from regular to chaotic.~I investigated the eigenvalue spacings in a system of~1 to 9 Dirac delta barriers placed in an infinite square well such that the ratio of the interval lengths between the barriers was irrational. I computed 1,000~energy eigenvalues of the sequence at three energy ranges: low~energy (the probability~that a particle is transmitted~through a delta barrier is close to zero), medium energy (the transmission probability~is~close to one half), and high energy (the~transmission probability~is~close to one).~I then unfolded the sequence, so that the average eigenvalue spacing was one, and~found the distribution of spacings.~For~six or more barriers~the low energy sequences followed~Poisson statistics, the medium energy sequences followed~GOE statistics, and~high energy sequences showed~Gaussian statistics peaked at one.~These results are interesting because this is~a very simple~system,~but~increasing the transmission probability shifts the statistics from Poisson to~GOE~to Gaussian.~ ~