Probability flow for a free particle: new quantum effects

Abstract

New quantum probability flow bounds and associated eigenfunctions are determined numerically for a free non-relativistic particle, with momentum lying in the range (p 0, ∞), for each fixed p 0 chosen in the range (−∞ , ∞). It is found that as p 0 increases through positive values, the maximum possible probability backflow from right-to left (R to L), opposite to the direction of all contributing momenta, decreases monotonically from the well-known value ≈0.038452 at p 0 = 0 but never reaches 0. As p 0 decreases through negative values, the maximum R to L flow increases monotonically but never reaches 1. These new quantum effects are compared and contrasted with the corresponding classical behaviour. A surprising new effect is revealed: Even when p 0 is negative, L to R directed momenta contribute to the maximum R to L flow. The size of this contribution is indicated by comparing with the maximum R to L flow possible when only negative momenta with values between p 0 and 0 are allowed. The extended modeling admits a simple interpretation of the classical limit without the introduction of a mechanism external to the system, as an effective value of Planck’s constant approaches zero.

Phase-space representation of diffraction in time: Analytic results

“…where the asterisk denotes complex conjugation. In view of (25) we hence refer to f as the Husimi amplitude. As is clear on (26), f is by construction the overlap between the transmitted state Ψ and a minimum-uncertainty Gaussian wave packet ψ σ, x, v (x, 0) of the form (1), hence having mean position x, mean velocity v and width σ.…”

“…x 0 , v 0 = t c , (D. 25) where we used the definition (11) of t c . Because t c > 0 by assumption, we deduce from (D.25) that the two stationary points τ…”

“…[23,24] for pioneering works on quantum backflow and the introduction section in Ref. [25] for the most up-to-date review of the literature).…”

On the experiment-friendly formulation of quantum backflow

“…Refs. [1,[4][5][6][7]. Thus, QB has been considered for a particle moving in the presence of linear [8] and short-range [9] potentials.…”

“…[4,9,17]. QB is also known to bear close relation to the arrival-time problem [2,3,[18][19][20][21][22][23][24][25] and some nonclassical aspects of the flow of probability in quantum systems [7,[26][27][28][29].…”