Dispersed Indeterminacy

TL;DR

This paper explores dispersed quantum states involving evanescent waves, challenging traditional views on phase space indeterminacy and showing that total momentum uncertainty can be finite despite large state dispersion.

Contribution

It introduces a model of dispersed states with a discrete Wigner function, analyzing their momentum indeterminacy and measurement limitations, thus expanding understanding of quantum indeterminacy.

Findings

States can have large total indeterminacy with separated phase space cells.

Momentum eigenstates for evanescent waves cannot be fully measured, limiting observed indeterminacy.

Total momentum uncertainty remains finite despite dispersed phase space representation.

Abstract

A state of a single particle can be represented by a quantum blob in the corresponding phase space, or by a cell in its 2-D subspace. Its area is frequently stated to be no less than one half of the Plank constant, implying that such a cell is an indivisible quantum of the 2-D phase space. But this is generally not true, as is evident, for instance, from representation of some states in the basis of innately discrete observables like angular momentum. Here we consider some dispersed states involving the evanescent waves (EW) different from that in the total internal reflection. Such states are represented by a set of separated point-like cells, but with a large total indeterminacy. An idealized model has a discrete Wigner function forming an infinite periodic array of dots on the phase plane. The question about the total momentum indeterminacy in such state is discussed. We argue that the transverse momentum eigenstates corresponding to the considered EW-s cannot be singled out by any known measurement procedure, and the whole infinite set of the corresponding eigenvalues can contribute only a certain fraction to the observed momentum indeterminacy which remains finite.