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The Philippine journal of science

Negative Probability Current in a Freely Falling Quantum Particle

Anthony Allan Villanueva

Published Jul 20, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.56899/150.05.25

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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.
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A wave packet 𝚿𝚿(𝒙𝒙, 𝒕𝒕) of a single particle has a statistical correlation between its position x and momentum p, quantified as the position-momentum covariance. The covariance influences wave packet spreading and the probability current through a given point. This paper shows another effect of the covariance: non-zero covariance can manifest as an asymmetry of the regional momentum density. Consider a selective measurement |𝚿𝚿⟩ → |𝚿𝚿[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] ⟩ where the initial state |𝚿𝚿⟩ is projected into the state |𝚿𝚿[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] ⟩ within a smaller region 𝒙𝒙𝟏𝟏 < 𝒙𝒙 < 𝒙𝒙𝟐𝟐. The momentum representation of the projected state is 𝚽𝚽[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] = ⟨𝒑𝒑|𝚿𝚿[𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] ⟩ and the corresponding momentum density |𝚽𝚽 [𝒙𝒙 𝟏𝟏 ,𝒙𝒙 𝟐𝟐 ] | 𝟐𝟐 is the regional momentum density. This paper examines a molecule-sized particle under uniform gravity (represented by a Gaussian wave packet 𝚼𝚼𝟎𝟎 ) at the classical turning point. I consider the effect of the covariance of 𝚼𝚼𝟎𝟎 on the regional momentum density in the classically forbidden region. The initial state 𝚼𝚼𝟎𝟎 with a given covariance is projected into the classically forbidden region, producing the state 𝚼𝚼𝑪𝑪𝑪𝑪. If the corresponding momentum wave function is 𝚽𝚽𝑪𝑪𝑪𝑪, the regional momentum density is 𝚷𝚷 𝑪𝑪𝑪𝑪 = |𝚽𝚽𝑪𝑪𝑪𝑪 |𝟐𝟐. I derive an analytic expression for 𝚷𝚷 𝑪𝑪𝑪𝑪 that shows that a non-zero covariance predicts an asymmetric momentum density in the classically forbidden region. This gives us a measure of control in preparing a preferred momentum distribution in the classically forbidden region using the appropriate covariance, at the price of a larger momentum uncertainty due to the uncertainty principle (as the configuration space of the particle is decreased). Also, since the momentum density is obtained experimentally from the statistics of momentum measurements, we can measure the covariance through comparison with the predicted momentum distribution, as well as indirectly test the equivalence principle.
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.