The free time of arrival problem in quantum phase space

Abstract

The theory of time of arrival (TOA) operators provides a general framework for incorporating the arrival time of a particle as a quantum observable within the rigged Hilbert space formulation of quantum mechanics. In this paper, we introduce a non-operator-based formalism by considering the arrival time problem of a free structureless particle in quantum phase space. We derive the corresponding arrival time expectation value and eigenvalue equation and show that our results correctly lead to the well-known predictions of the theory of TOA operators. This paper then offers a possibility of developing an alternative formalism that resolves quantum time of arrival problems using only c-numbers and real-valued functions instead of non-commuting operators.