Time of arrival in quantum mechanics

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. © 1996 The American Physical Society.