2025/11/04 by Magadan, John Jaykel P., Galapon, Eric A.
Computer Science · Physics and Astronomy · #FOS: Physical sciences #Quantum Information and Cryptography #Quantum Physics (quant-ph) #Quantum chaos and dynamical systems #stochastic dynamics and bifurcation
paper · doi:10.48550/arxiv.2511.02307
openalex publication_date 2025/11/04 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28
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.