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American Journal of Physics

The negative flow of probability

Anthony Allan D. Villanueva

Published Mar 19, 2020• 6 citations
Publisher

Abstract

We derive an inequality that the parameters of a 1D free-particle Gaussian wave packet with a positive group velocity, approaching a given region x > q, must satisfy such that a negative probability current J exists on q. Local probability conservation implies the counter-intuitive result that the particle detection probability in the region x > q is actually decreasing. The condition J < 0 requires the negative correlation of the position and momentum observables of the state, but the time scales for the negative current and anti-correlation regimes are not identical. Using a probability current operator, we obtain an integral representation of J in momentum space for any free particle wave packet. We use this integral representation to distinguish the separate contributions to J by the positive and negative momentum components, and we identify a third contribution to J composed of cross-terms of both momenta. For the specific case of a Gaussian wave packet with a negative correlation between its position and momentum, the positive momentum component can contribute a negative value to the probability current.

DOI
10.1119/10.0000856

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We derive an inequality that the parameters of a 1D free-particle Gaussian wave packet with a positive group velocity, approaching a given region x > q, must satisfy such that a negative probability current J exists on q. Local probability conservation implies the counter-intuitive result that the particle detection probability in the region x > q is actually decreasing. The condition J < 0 requires the negative correlation of the position and momentum observables of the state, but the time scales for the negative current and anti-correlation regimes are not identical. Using a probability current operator, we obtain an integral representation of J in momentum space for any free particle wave packet. We use this integral representation to distinguish the separate contributions to J by the positive and negative momentum components, and we identify a third contribution to J composed of cross-terms of both momenta. For the specific case of a Gaussian wave packet with a negative correlation between its position and momentum, the positive momentum component can contribute a negative value to the probability current.
A quantum particle (represented by a Gaussian wave packet) under uniform gravity, projected upward (the conventional positive direction) along the vertical degree of freedom x, may have a negative probability current on a set of points q. This imposes the condition that the position and momentum observables of the state are negatively correlated. We calculate the effect of the negative correlation on the individual contributions to the probability current by the positive and negative momentum amplitudes of the state using an integral representation of the probability current. Negative position-momentum correlation causes a transient attenuation of all contributions to the probability flow, and wave packet spreading is temporarily mitigated.
We discuss a class of solutions of the time-dependent Schrödinger equation such that the position uncertainty temporarily decreases. This self-focusing or contractive behavior is a consequence of the anti-correlation of the position and momentum observables. Since the associated position density satisfies a continuity equation, upon contraction the probability current at a given fixed point may flow in the opposite direction of the group velocity of the wave packet. For definiteness, we consider a free particle incident from the left of the origin, and establish a condition for the initial position-momentum correlation such that a negative probability current at the origin is possible. This implies a decrease in the particle's detection probability in the region x > 0, and we calculate how long this occurs. Analogous results are obtained for a particle subject to a uniform gravitational force if we consider the particle approaching the turning point. We show that position-momentum anti-correlation may cause a negative probability current at the turning point, leading to a temporary decrease in the particle's detection probability in the classically forbidden region.
Pure states of a free particle in non-relativistic quantum mechanics are described, in which the probability of finding the particle to have a negative x-coordinate increases over an arbitrarily long, but finite, time interval, even though the x-component of the particle's velocity is certainly positive throughout that time interval. It is shown that, for any state of this type, the greatest amount of probability which can flow back from positive to negative x-values in this counter-intuitive way, over any given time interval, is equal to the largest eigenvalue of a certain Hermitian operator, and it is estimated numerically to have a value near 0.04. This value is not only independent of the length of the time interval and the mass of the particle, but is also independent of the value of Planck's constant. It reflects the structure of Schrodinger's equation, rather than the values of the parameters appearing there. Backflow of positive probability is related to the non-positivity of Wigner's density function, and can be regarded as arising from a flow of negative probability in the same direction as the velocity. Generalizations are indicated, to the relativistic free electron, and to non-relativistic cases in which probability backflow occurs even in opposition to an arbitrarily strong constant force.
We present an introduction to the backflow effect in quantum mechanics – the phenomenon in which a state consisting entirely of positive momenta may have negative current and the probability flows in the opposite direction to the momentum. We show that the effect is present even for simple states consisting of superpositions of gaussian wave packets, although the size of the effect is small. Inspired by the numerical results of Penz et al, we present a wave function whose current at any time may be computed analytically and which has periods of significant backflow, with a backwards flux equal to about 70 percent of the maximum possible backflow, a dimensionless number c bm ≈ 0.04, discovered by Bracken and Melloy. This number has the unusual property of being independent of ℏ (and also of all other parameters of the model), despite corresponding to a quantum-mechanical effect, and we shed some light on this surprising property by considering the classical limit of backflow. We conclude by discussing a specific measurement model in which backflow may be identified in certain measurable probabilities.
A free particle represented by a Gaussian wave packet with a negative statistical correlation between the position and momentum can have a decreasing position uncertainty (or wave packet contraction) for a finite duration. We numerically simulate particle scattering in such a Gaussian state with a step potential barrier to see if the contraction is still present. The time-dependent Schrödinger equation (TDSE) was solved using the Crank-Nicolson method implemented in the Python programming language. We show that with a negative position-momentum correlation, the incident Gaussian wave packet and the reflected wave packet both exhibit a persistent contractive behavior. This particle localization suggests that more precise position measurements during barrier scattering events beyond the standard quantum limit are possible.
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.