Classical and quantum spreading of position probability

Abstract

If Δxt is the standard deviation of the position probability distribution at time t of a particle moving freely in one dimension, and Δv is the standard deviation of its velocity distribution, it is shown for a suitable choice of time origin that (Δxt)2 = (Δx0)2 + (Δv)2t2 whether the particle is moving classically or quantum mechanically. The quantum-mechanical proof is shown to exactly parallel that in the classical case. In the classical case, the correlation coefficient between the velocity and position distributions is shown to tend to −1 as t → −∞, to increase steadily with time, and to become +1 as t → +∞. A quantum-mechanical analog of this result is also proved.