The Effect of Spatial Disorder on Eigenvalue Statistics and Eigenstate Structure in a Simple Quantum System

We examine the effect of introducing spatial disorder on the energy eigenvalue statistics and eigenstate structure for a particle in an infinite square well of width L with twelve Dirac delta barriers placed inside. When the barriers are placed at regular intervals the distribution of spacings does not match any standard distribution and the eigenstates are generally delocalized. Spatial disorder is introduced through random barrier displacements drawn from a Gaussian distribution with mean zero and standard deviation \(\sigma L\). As \(\sigma \) is increased the system becomes disordered and the resulting level spacing distribution depends on the transmission probability T through each barrier: Poisson-like for \(T\approx 0\), a Brody distribution for \(T=0.5\), a Wigner GOE distribution for \(T\approx 0.7\), and Gaussian for \(T\approx 1\). The transition in the level spacing statistics takes place over a range of approximately \(10^{-4}< \sigma < 10^{-3}\) in all cases, with the reduced chi-square values for the fit to the relevant distribution following a power law in \(\sigma \) within the transition range. These results show that even a small degree of spatial disorder (two orders of magnitude smaller than the distance between barriers) is sufficient to produce eigenvalue statistics that match the limiting distribution for the highly disordered system. In addition, as disorder is increased the eigenstates become strongly localized for \(T\approx 0\), but remain delocalized for \(T\approx 1\) and show only weak localization at intermediate values of T.