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American Journal of Physics

Self-focusing quantum states

Anthony Allan D. Villanueva

Published Jan 18, 2018• 1 citation
Publisher

Abstract

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.

DOI
10.1119/1.5009921

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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.
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.
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.
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.
We consider the time evolution of a special class of wave packets representing the motion of a free particle along a line. In order to make contact with an earlier work on s-waves in two or higher dimensions we confine ourselves to wave packets with a probability density that is symmetric with respect to the origin. Here we concentrate on the phenomenon that an appropriately prepared wave packet is focusing before it displays its familiar behavior of spreading. We derive the condition on the wave packet for this effect to occur: the probability density at the origin has to be concave. The translation of this requirement into Wigner phase space reads: the average of the kinetic energy along the momentum axis has to be negative. In this way we identify the origin of this focusing effect of the wave packet as quantum interference.
We analyze the time-behaviour of the Gaussian wave packet of a quantum particle moving in the field of a linear force, undergoing a continuous nondemolition observation of its position. In particular, the time-dependence of position and momentum dispersions, as well as position-momentum uncertainty is investigated. The discussion includes the case of a free particle and initially coherent state of a harmonic oscillator. It is found that in the beginning of observation the momentum dispersion and position-momentum uncertainty always increase in contrast to the position dispersion. It is also proved that all considered functions asymptotically take the form of harmonic oscillations, decaying to finite values. For initially coherent states, the asymptotic bounds for all considered functions are given.
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.