TIME OF ARRIVAL IN QUANTUM-MECHANICS

We study the problem of computing the probability for the time of arri val of a quantum particle at a given spatial position. We consider a s olution to this problem based on the spectral decomposition of the par ticle'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 explici tly in the simple case of a free nonrelativistic particle and compare the probabilities it yields with the ones estimated indirectly in term s of the flux of the Schrodinger current. We derive a well-defined unc ertainty relation between time of arrival and energy; this result show s that the well-known arguments against the existence of such a relati on can be circumvented. Finally, we define a ''time representation'' o f the quantum mechanics of a free particle, in which the time of arriv al is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between inde pendent (time) and dependent (position) variables, analogous to the on e revealed by the parametrized formalism in classical mechanics.