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arXiv:1608.07644

Remarks on Quantum Probability Backflow

Anthony J. Bracken, James McGuire

Published Aug 27, 2016• 1 citation
arXivHTMLPDF

Abstract

It is known that for a non-relativistic quantum particle traveling freely on the $x$-axis, the positional probability can flow in the opposite direction to the particle's velocity. The maximum possible amount of such backflow that can occur over any time interval has been determined previously as the largest positive eigenvalue of a certain hermitian observable, with the value $0.0384517\dots$, or about $4\%$ of the total probability on the line. The eigenvalue problem is now considered numerically in the more general case of states with momentum restricted to the range $p_0

DOI
10.48550/arxiv.1608.07644

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It is known that for a non-relativistic quantum particle traveling freely on the xxx-axis, the positional probability can flow in the opposite direction to the particle's velocity. The maximum possible amount of such backflow that can occur over any time interval has been determined previously as the largest positive eigenvalue of a certain hermitian observable, with the value 0.0384517…0.0384517\dots0.0384517…, or about 4%4\%4% of the total probability on the line. The eigenvalue problem is now considered numerically in the more general case of states with momentum restricted to the range $p_0
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.
Abstract In his study of the ‘time of arrival’ problem in the nonrelativistic quantum mechanics of a single particle, Allcock [1] noted that the direction of the probability flux vector is not necessarily the same as that of the mean momentum of a wave packet, even when the packet is composed entirely of plane waves with a common direction of momentum. Packets can be constructed, for example for a particle moving under a constant force, in which probability flows for a finite time in the opposite direction to the momentum. A similar phenomenon occurs for the Dirac electron. The maximum amount of probabilitiy backflow which can occur over a given time interval can be calculated in each case.
In his study of the ‘time of arrival’ problem in the nonrelativistic quantum mechanics of a single particle, Allcock [1] noted that the direction of the probability flux vector is not necessarily the same as that of the mean momentum of a wave packet, even when the packet is composed entirely of plane waves with a common direction of momentum. Packets can be constructed, for example for a particle moving under a constant force, in which probability flows for a finite time in the opposite direction to the momentum. A similar phenomenon occurs for the Dirac electron. The maximum amount of probabilitiy backflow which can occur over a given time interval can be calculated in each case.
We investigate the backflow effect in elementary 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 compute the current and flux for states consisting of superpositions of Gaussian wave packets. These are experimentally realizable but the amount of backflow is small. Inspired by the numerical results of Penz et al. [Penz, Gr\"ubl, Kreidl, and Wagner, J. Phys. A 39, 423 (2006)], we find two nontrivial wave functions whose current at any time may be computed analytically and which have periods of significant backflow, in one case with a backward flux equal to about 70% of the maximum possible backflow, a dimensionless number ${c}_{bm}\ensuremath{\approx}0.04$, discovered by Bracken and Melloy [Bracken and Melloy, J. Phys. A 27, 2197 (1994)]. This number has the unusual property of being independent of $\ensuremath{\hbar}$ (and also of all other parameters of the model), despite corresponding to an obviously quantum-mechanical effect, and we shed some light on this surprising property by considering the classical limit of backflow. We discuss some specific measurement models in which backflow may be identified in certain measurable probabilities.
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.
Measurable quantities that have positive values in classical dynamical systems need not to be positive in quantum theory. For example, consider a free quantum mechanical particle in one dimension. There are quantum states in which the particle's velocity is positive with probability 1, but where the probability flux for its position is locally negative; that is, while its velocity points to the right, the particle travels to the left. These effects are however small and limited in space and time by certain lower bounds, which are called "quantum inequalities". Similar effects also appear for a particle whose motion is governed by a Schroedinger equation with a certain class of potentials.