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arXiv:2505.13184 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Repeated quantum backflow and overflow

Christopher J. Fewster, Harkan J. Kirk-Karakaya

Published Dec 15, 2025• 2 citations
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Abstract

Abstract Quantum particles moving in one dimension with rightwards momentum can exhibit the surprising phenomenon of quantum backflow (QB): a net probability transfer to the left-hand half-line over a finite time interval. We generalize this phenomenon by considering the sum of probability differences for M disjoint time intervals. In classical mechanics, the total backflow lies in the interval [−1,0] for all M, indicating rightwards probability transfer. By contrast, we show that the maximum M-fold QB is positive and unbounded from above as M increases, demonstrating the existence of repeated backflow. For M≥2, a new phenomenon of ‘quantum overflow’ is discovered: there are states whose total backflow is below −1, which is impossible for classical particles. The extent of the backflow and overflow effects is described by a hierarchy of backflow and overflow functions and constants, of which the M=1 backflow constant was first studied by Bracken and Melloy. Limiting cases of the backflow and overflow functions are studied, including cases in which two disjoint intervals merge. The analytical results are supported by detailed numerical investigations. Using numerical acceleration methods, we obtain a new estimate of the Bracken–Melloy constant of 0.0384506, slightly lower than the previously accepted value of 0.038452.

DOI
10.1098/rspa.2025.0577

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Abstract Quantum particles moving in one dimension with rightwards momentum can exhibit the surprising phenomenon of quantum backflow (QB): a net probability transfer to the left-hand half-line over a finite time interval. We generalize this phenomenon by considering the sum of probability differences for M disjoint time intervals. In classical mechanics, the total backflow lies in the interval [−1,0] for all M, indicating rightwards probability transfer. By contrast, we show that the maximum M-fold QB is positive and unbounded from above as M increases, demonstrating the existence of repeated backflow. For M≥2, a new phenomenon of ‘quantum overflow’ is discovered: there are states whose total backflow is below −1, which is impossible for classical particles. The extent of the backflow and overflow effects is described by a hierarchy of backflow and overflow functions and constants, of which the M=1 backflow constant was first studied by Bracken and Melloy. Limiting cases of the backflow and overflow functions are studied, including cases in which two disjoint intervals merge. The analytical results are supported by detailed numerical investigations. Using numerical acceleration methods, we obtain a new estimate of the Bracken–Melloy constant of 0.0384506, slightly lower than the previously accepted value of 0.038452.
It has been known since at least 1969, that there exist quantum states of a free particle on the line with purely positive momentum that exhibit probability transfer towards the negative half-line. This phenomenon, the quantum backflow effect, is the subject of this thesis. The largest amount of backflow any state can exhibit over a single time interval, the Bracken–Melloy constant, was initially calculated to be cBM ≈ 0.04. In the first half of this thesis, we consider the question of how much quantum backflow states can exhibit over disjoint time intervals. This is done by formulating the question in terms of the spectra of bounded operators. When considering more than one disjoint time interval, we discuss a new phenomenon where a state exhibits more probability transfer in the same direction as its momentum than any classical state. We call this effect quantum overflow. Given a number M of disjoint time intervals, we give bounds on the maximum amount of backflow and overflow a state can exhibit over any M many intervals. The limiting cases in which two disjoint intervals merge into one is particularly studied. Finally, we show plots of the time t = 0 momentum space wavefunction of states exhibiting multiple quantum backflow and quantum overflow. The second half of the thesis contains a detailed numerical investigation into the single backflow problem. Based on previous conjectures, we construct the maximum backflow state from a given set of normalised states closely resembling the sinc function. The Bracken—Melloy constant is then bounded from below by the largest solution to a generalised eigenvalue problem. Using a multi-step numerical procedure, we solve a high precision large dense generalised eigenvalue problem. From this we obtain new numerical approximation to the backflow state and the best known rigorous lower bound of the Bracken—Melloy constant.
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.
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.
Abstract. We derive some rigorous results concerning the backflow operator introduced by Bracken and Melloy. We show that it is linear bounded, self adjoint, and not compact. Thus the question is underlined whether the backflow constant is an eigenvalue of the backflow operator. From the position representation of the backflow operator we obtain a more efficient method to determine the backflow constant. Finally, detailed position probability flow properties of a numerical approximation to the (perhaps improper) wave function of maximal backflow are displayed. PACS numbers: 03.65 1. Introduction and
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.