Fractality of Certain Quantum States

Abstract

Fractal structures appearing in solutions of certain quantum problems are investigated. We prove the previously announced results concerning the existence and properties of fractal states for the Schrödinger equation in the infinite one-dimensional well. In particular, we show that for this problem, there exist solutions in the form of fractal quantum carpets: the probability density P(x,t) forms a fractal surface with dimension D_xy, while its cross-sections P_t(x) and P_x(t) typically form fractal graphs with dimensions D_x and D_t, respectively, where D_xy=2+D_x/2 and D_t=1+D_x/2 (almost everywhere).