Surprises in the suddenly-expanded infinite well

Abstract

I study the time evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density |Ψ(x, t)|2 shows up quite a surprising behaviour: for definite times, plateaux appear for which |Ψ(x, t)|2 is constant on finite intervals for x. Elements of theoretical explanation are given by analysing the singular component of the second derivative ∂xxΨ(x, t). Analytical closed expressions are obtained for some specific times, which easily allow us to show that, at these times, the density organizes itself into regular patterns provided the size of the box is large enough; more, above some critical size depending on the specific time, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.