The Schrödinger density and the Talbot effect
Volume 72 / 2006
Abstract
We study the local
properties of the time-dependent probability density function for
the free quantum particle in a box, i.e. the squared magnitude of
the solution of the Cauchy initial value problem for the
Schrödinger equation with zero potential, and the periodic
initial data. $\sqrt{\delta}\,$-families of initial functions
are considered whose squared magnitudes approximate the periodic
Dirac $\delta$-function. The focus is on the set of rectilinear
domains where the density has a special character, in particular,
remains bounded, or even has low average values (“the valleys of
shadows”).
An essential part of the paper is dedicated to a review of some
earlier results concerning the fractal properties of
Vinogradov's extensions, which incorporate the solutions of a wide
class of Schrödinger type equations. Relations are discussed
with the optical diffraction phenomena discovered in 1836 by
W. H. F. Talbot, the British inventor of photography. In the modern
Physics literature, self-similarity in the wave diffracted by
periodic gratings, is known as fractional and fractal
revivals, and quantum carpets (M. Berry, W. Schleich, and
many others). Self-similarity has been well-known, and
extensively utilized in Analytic Number Theory, since the creation
of the circle method of Hardy–Littlewood–Ramanujan, and
Vinogradov's method of estimation and asymptotic formulas for H.
Weyl's exponential sums. According to these methods, on the
major arcs}, the complete rational exponential sums are the {\em
scaling factors, while the appropriate oscillatory integrals
constitute the pattern of the arising arithmetical
carpets.
