Dirac delta-convergence of free-motion time-of-arrival eigenfunctions

TL;DR

This paper mathematically proves that eigenfunctions of the quantum time-of-arrival operator, modeled by the Aharonov-Bohm operator, converge to a delta function at the arrival time, confirming their interpretation as states with definite arrival time.

Contribution

It provides a rigorous proof that the eigenfunctions of the AB operator with complex eigenvalues evolve to collapse at the arrival point, establishing their physical significance.

Findings

Eigenfunctions with complex eigenvalues evolve unitarily to collapse at the arrival point.

Time-evolved probability density forms a delta sequence as the imaginary part of eigenvalues approaches zero.

Mathematical validation of the quantum time-of-arrival eigenfunctions' interpretation.

Abstract

Previous numerical analyses on the Aharonov-Bohm (AB) operator representing the quantum time-of-arrival (TOA) observable for the free particle have indicated that its eigenfunctions represent quantum states with definite arrival time at the arrival point. In this paper, we give the mathematical proof that this is indeed the case. An essential element of this proof is the consideration of the eigenfunctions of the AB operator with complex eigenvalues. These eigenfunctions can be considered legitimate TOA eigenfunctions because they evolve unitarily to collapse at the arrival point at the time equal to the real part of their eigenvalue. We show that the time-evolved TOA position probability density distribution evaluated at the time equal to the real part of the eigenvalue forms a dirac delta sequence in the limit as the imaginary part of the eigenvalue approaches zero.