Phase Space Evolution and Discontinuous Schr\"odinger Waves

TL;DR

This paper investigates how discontinuous wavefunctions evolve in phase space, revealing affine transformations that generate diffraction patterns and constant probability trajectories, offering new insights into quantum wave propagation.

Contribution

It introduces a novel phase space perspective on Schrödinger evolution of discontinuous waves, linking affine transformations to diffraction phenomena and trajectory formation.

Findings

Affine transformations cause diffraction patterns.

Infinite space-time trajectories emerge from wave discontinuities.

SL(2) evolution map produces constant probability trajectories.

Abstract

The problem of Schr\"odinger propagation of a discontinuous wavefunction -diffraction in time- is studied under a new light. It is shown that the evolution map in phase space induces a set of affine transformations on discontinuous wavepackets, generating expansions similar to those of wavelet analysis. Such transformations are identified as the cause for the infinitesimal details in diffraction patterns. A simple case of an evolution map, such as SL(2) in a two-dimensional phase space, is shown to produce an infinite set of space-time trajectories of constant probability. The trajectories emerge from a breaking point of the initial wave.