Time of arrival in quantum mechanics

Abstract

We study the problem of computing the probability for the time-of-arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the "time-of-arrival" operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle, and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schrödinger current. We derive a well defined uncertainty relation between time-of-arrival and energy; this result shows that the well known arguments against the existence of such a relation can be circumvented. Finally, we define a "time-representation" of the quantum mechanics of a free particle, in which the time-of-arrival is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between independent (time) and dependent (position) variables, analogous to the one revealed by the parametrized formalism in classical mechanics.

Space-time-symmetric extension of nonrelativistic quantum mechanics

“…This is exactly the time-of-arrival probability density obtained by several authors via different approaches [8,[30][31][32][33] and is in excellent agreement with numerical "quantum jump" time-of-flight simulations [18]. However, the models employed have faced problems even in the free particle case.…”

Weak measurement of arrival time

“…The arrival time distribution may be useful solving the tunneling time problem, as well. Therefore, the quantum description of the arrival time has attracted much attention [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].…”

Arrival time distributions and perfect absorption in classical and quantum mechanics

“…The former distribution was originally derived by imposing a series of conditions consistent with the classical distribution [17], and has been later studied, rederived or generalized by several authors [2,[5][6][7][8][9][10][11][12] [The transformation (11) does not hold when J is replaced by Π K because of the non linearity introduced by the square root in (12)]. It arises naturally as the square of the overlap between the initial state (restricted to positive momenta) and the eigenstates of the "time of arrival operator"…”

“…Despite this there are experiments, most notably "time of flight experiments", that seemingly circumvent the theoretical objections and difficulties exemplified by Allcock's work [1]. Several researchers have tried in recent years to fill this gap between theory and practice by reexamining the subject using a variety of approaches [2][3][4][5][6][7][8][9][10][11][12][13][14]; a recent review provides a brief summary of the methods applied and a discussion of open questions [14].…”

“…We may first consider an incident state |ψ that leads to analytical expressions for x|ψ(t) and J(x, t), and that satisfies the stated restrictions of the arrival time theories of references [17,5,9,10,12],…”