Contraction of a Wave Packet while Scattering with a Step Potential
Vincent Gene L. Otero and Anthony Allan D. Villanueva*
Institute of Physics, University of the Philippines Los Baños,
College, Los Baños, Laguna 4031 the Philippines
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.
INTRODUCTION
In quantum mechanics, the wave property of a microscopic particle (10–9 m or smaller) manifests when it encounters a potential barrier. Mathematically, the particle is modeled as a wave packet in position-space. This assumes that the particle momentum is no longer definite. Instead, there is a spread of momenta of order ∆𝑝 (the momentum uncertainty) representing the coherent superposition of different momentum states. The dispersion and propagation of this wave packet are determined by the time-dependent Schrödinger . . . . read more
