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arXiv:1902.07952 · American Journal of Physics

Unusual broadening of wave packets on lattices

Kurt Schönhammer

Published Feb 19, 2019• 9 citations
arXivHTMLPDF

Abstract

The broadening of one-dimensional Gaussian wave packets is presented in all textbooks on quantum mechanics. It is used as an example to elucidate Heisenberg's uncertainty relation. The behaviour on a lattice is drastically different if the amplitude and/or phase of the wave packet varies on the scale of the lattice constant. Important examples are very narrow wavepackets or wavepackets with an average velocity comparable to the maximum velocity on the lattice. The analytical and numerical results for the time dependence of wave packets on a one-dimensional lattice are presented.

DOI
10.1119/1.5089752

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The broadening of one-dimensional Gaussian wave packets is presented in all textbooks on quantum mechanics. It is used as an example to elucidate Heisenberg's uncertainty relation. The behaviour on a lattice is drastically different if the amplitude and/or phase of the wave packet varies on the scale of the lattice constant. Important examples are very narrow wavepackets or wavepackets with an average velocity comparable to the maximum velocity on the lattice. The analytical and numerical results for the time dependence of wave packets on a one-dimensional lattice are presented.
The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.
The usual textbook discussions of the velocity and spreading of a wave packet are based on special cases and approximations which have a limited range of validity. In this paper more general relations are obtained by considering the time dependence of the moments of a wave packet. Both a packet of real sinusoidal plane waves (a real pulse, e.g., sound) and a packet of complex exponential plane waves (e.g., the quantum-mechanical wave function of a free particle) are considered. The results are similar if the real wave packet contains no standing wave components. It is shown that the centroid of a wave packet moves at a constant velocity which is a weighted average of the group velocity. The spreading of the wave packet depends on the product of the variance of the group velocity and the square of the time interval before or after the time of minimum packet width. The packet moves and spreads like a collection of particles in free flight, in which the velocities of the particles are the group velocities of the wave components of the packet.
The wave function temporal evolution on the one-dimensional (1D) lattice is considered in the tight-binding approximation.The lattice consists of N equal sites and one impurity site (donor).The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor.The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2.The wave packet width grows with time ∼ t 1/3 and its amplitude decreases ∼ t -1/3 .The wave packet reflects multiply from the lattice ends.Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.