User:Yearsley/time

Unlike in the classical theory, in Quantum mechanics there is no unique prescription for calculating the time of arrival of a wave packet at a given region of space. Different interpretations of quantum theory give rise to differing expressions for the arrival time distribution, and understanding the origin of these differences is a major challenge in the foundations of quantum mechanics.

Arrival time in classical mechanics
Consider an ensemble of free, non-interacting particles in 1-d described by a phase space distribution $$F(q,p,t)$$ normalised to one, that satisfies $$F(q,p/leq 0,t)=0$$, so that all the particles are moving to the right. For free motion each particle with initial position and momentum $$(q_0,p_0)$$ crosses the origin at a time given by $$T=-q_0 m/p_0 $$and the distribution of arrival times, $$/Pi_0(T) $$, is given by the probability current at the origin $$ J_0(T) $$