Scattering Problems via Real-time Wave Packet Scattering

TL;DR

This paper introduces a simple numerical wave packet scattering method for one-dimensional lattice models, providing an intuitive physical picture and accessible diagonalization approach, with results benchmarked against analytic solutions.

Contribution

It presents a straightforward, educational numerical method for lattice scattering problems using wave packets and matrix diagonalization, making quantum scattering more accessible to students.

Findings

Numerical results match analytic solutions for scattering in lattice models.

Identification of resonant wave packet momenta in a dimer system.

Comparison of lattice and continuum scattering scenarios.

Abstract

In this paper, we use a straightforward numerical method to solve scattering models in one-dimensional lattices based on a tight-binding band structure. We do this by using the wave packet approach to scattering, which presents a more intuitive physical picture than the traditional plane wave approach. Moreover, a general matrix diagonalization method that is easily accessible to undergraduate students taking a first course in quantum mechanics is used. Beginning with a brief review of wave packet transport in the continuum limit, comparisons are made with its counterpart in a lattice. The numerical results obtained through the diagonalization method are then benchmarked against analytic results. The case of a resonant dimer is investigated in the lattice, and several resonant values of the mean wave packet momentum are identified. The transmission coefficients obtained for a plane wave incident on a step potential and rectangular barrier are compared by investigating an equivalent scenario in a lattice. Lastly, we present several short simulations of the scattering process which emphasize how a simple methodology can be used to visualize some remarkable phenomena.