Quantum flux and reverse engineering of quantum wave functions

Abstract

An interpretation of the probability flux is given, based on a derivation of its eigenstates and relating them to coherent state projections on a quantum wavefunction. An extended definition of the flux operator is obtained using coherent states. We present a "processed Husimi" representation, which makes decisions using many Husimi projections at each location. The processed Husimi representation reverse engineers or deconstructs the wavefunction, yielding the underlying classical ray structure. Our approach makes possible interpreting the dynamics of systems where the probability flux is uniformly zero or strongly misleading. The new technique is demonstrated by the calculation of particle flow maps of the classical dynamics underlying a quantum wavefunction in simple model systems such as a circular billiard with and without a magnetic field.

Five momenta

“…It is clear that, as stated earlier, k(r) carries only part of the momentum distribution at the point r. A complete description is contained in the Wigner function or any of its relatives, such as the Husimi. The latter has recently been processed in an imaginative way [29] to yield a pictorial representation of a series of local momenta r, related in semiclassical cases to the momenta of classical trajectories through r in cases where there are several.…”

Revealing the flux: Using processed Husimi maps to visualize dynamics of bound systems and mesoscopic transport

“…The first approach utilizes the algorithm outlined in Mason et al 1 , which simplifies the full Husimi projection at each point, to give rise to the significant classical paths within the system. The second approach, which we describe in Section III C, identifies key points where the system boundary deflects straight classical paths, giving rise to the structure of the quantum wavefunction.…”

“…In Mason et al 1 , we introduced the "processed Husimi map", which extends the concept of the probability flux, or probability current j (r, p), defined aŝ j r = 1 2m (|r r|p +p |r r|) ,…”

“…Husimi projections sampled across a system produce a Husimi map. By processing Husimi maps using the methods outlined in Mason et al 1 , we can show the locations and directions of classical trajectories suggested by a wave function. This technique has been used to examine the relationship between graphene boundary types and the classical dynamics of quasiparticles in each valley of the honeycomb dispersion relation, studying states with energies both close to and far from the Dirac point 4 .…”

“…This technique has been used to examine the relationship between graphene boundary types and the classical dynamics of quasiparticles in each valley of the honeycomb dispersion relation, studying states with energies both close to and far from the Dirac point 4 . Although the processed Husumi approach was presented in summary before 1 , the level of detail might be insufficient to allow non-experts to develop their own programs. Here, we also introduce new map-pings that quantify how boundaries and external fields affect those trajectories.…”