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The Philippine journal of science

Contraction of a Wave Packet while Scattering with a Step Potential

Vincent Gene L. Otero, Anthony Allan Villanueva

Published Feb 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.56899/154.01.20

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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 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.
The scattering process of wave packets is often described recovering the simple monochromatic scheme supposing that the particle comes from −∞ with a sharp momentum distribution, thus allowing for approximations usually used in common calculations of physical quantities involved in the process. In this work, we study the scattering process of a Gaussian wave packet impinging on the step potential, finding out that its dynamics depends on the initial conditions of the incident particle, and in particular on the wave packet origin x0. We propose a semi-classical approximated model to describe the dynamics of the scattering wave packet, also defining a characteristic time interval tf as the time required for the formation of transmitted packet beyond the step. Through a comparison with the numerical solution of the SchrÖdinger equation, our model explains the tf as a function of the origin x0 and spread of the incident wave packet in coordinate space, giving rise to such dependence for a finite distance scattering processes.
We consider the time development of a one-dimensional Gaussian wave packet. Noting that the quantum mechanical predictions concerning position and momentum measurements of a free particle described by a Gaussian packet are consistent with the predictions one would make on the basis of a classical “Naive Model,” we investigate whether this model can account for the scattering of a particle by a rectangular barrier. Obtaining numerical solutions of the time dependent Schrödinger equation for a series of packets with varying incident momenta, we find that the Naive Model is inadequate to explain the quantum mechanical transmission. We study the relative speeds of the transmitted and reflected packets, discussing a number of possible interpretations and indicating the quantum mechanical result. Finally, we note that for certain choices of the parameters describing the packet and the barrier, the reflected packet is composed of two distinct humps.
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 simple exact solution for the spacetime particle propagator in the presence of a one-dimensional rectangular potential is obtained. This propagator solves the Cauchy problem for the corresponding time-dependent Schrodinger equation in terms of the forward-moving (positive momentum) and backward-moving (negative momentum) components of the wavefunction. The exact formulae for the time-dependent wavefunction in different spatial regions (before, inside and after the barrier) are found. These results demonstrate that generally the forward- and backward-moving components of the wavefunction as well as their interference contribute to the probability density of finding a particle in different spacetime regions. The time of arrival to and the dwell time in the above-mentioned spatial regions are also obtained. The relative contribution of the positive- and negative-momentum components of the wavefunction as well as those of the below- and above-barrier energy regions are discussed for the case of an initial Gaussian wave packet. Numerical visualization of the probability density to find a particle in the different spacetime regions is performed and discussed in the tunneling regime.
This paper presents a robust computational study of the Time-Dependent Schrödinger Equation (TDSE) dynamics for a spatial Gaussian wave packet confined within a one-dimensional infinite potential well. Employing a discrete Finite Difference Method (FDM) framework, we construct a tridiagonal Hamiltonian matrix to compute the unitary real-time evolution of the system. We systematically analyze two critical quantum phenomena: (1) scattering and quantum tunneling against an engineered central rectangular potential barrier, and (2) temporal dispersion and subsequent fractional quantum revival of the localized state. The numerical solver successfully resolved a distinct fractional revival and constructive self-interference peak at t = 1.416 time units. These computational benchmarks provide a rigorous pedagogical and diagnostic baseline for grid-based matrix methods in low-dimensional quantum mechanics.