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Theoretical and Mathematical Physics

Integral Characteristics of Wave Packets in the Problem of the Evolution of A Wave Function on A One-Dimensional Lattice

V. N. Likhachov, George A Vinogradov

Published Nov 1, 2018• 0 citations
Publisher

Abstract

We consider the quantum dynamics of charge transfer on a lattice in the tight-binding approximation and analytically calculate the integral characteristics of the wave packet propagating along the lattice. We focus on calculating the mean and root-mean-square displacements. We also obtain expressions for higher-order moments as series for squares of Bessel functions, which might be independently interesting.

DOI
10.1134/s0040577918110053

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We consider the quantum dynamics of charge transfer on a lattice in the tight-binding approximation and analytically calculate the integral characteristics of the wave packet propagating along the lattice. We focus on calculating the mean and root-mean-square displacements. We also obtain expressions for higher-order moments as series for squares of Bessel functions, which might be independently interesting.
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.
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 ～ t1/3and 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.
In this work, we develop a time-dependent mathematical model to investigate the wave packet evolution of both massive and massless Dirac fermions in a tight-binding lattice. Our study focuses on the quantum diffusion of carriers in a one-dimensional crystal, where the system Hamiltonian is represented by both the Dirac and Schrödinger equations. Interestingly, during the quantum diffusion process, we observe oscillations of the wave packet between the spinor-up and spinor-down bands in the Dirac case. In the massive fermion scenario, these oscillations between the sublattices are influenced by inertial mass effects, resulting in longer oscillation periods as the mass increases. In addition, we find a strong dependence of Dirac fermion dynamics on the Fermi velocity in the tight-binding chain.
For a quantum particle on a one-dimensional lattice, the dynamics under the influence of a time-dependent, white-noise potential is shown to differ significantly from the classical diffusive behavior. In particular, the square of the distance displaced from origin is proportional to t0.5−−0.6{\mathit{t}}^{0.5--0.6}t0.5−−0.6. The origin of this dispersive behavior is traced to the special properties of the force-force correlation function, which integrates to zero and has a magnitude varying as ${\mathit{t}}^{\mathrm{\ensuremath{-}}0.5}$. These characteristics are attributed to the continuous spread of the quantum wave packet as well as its interaction with the discrete lattice.
In the “tight-binding” approximation (the Hückel model), we consider the evolution of the charge wave function on a semi-infinite one-dimensional lattice with an additional energy U at a single impurity site. In the case of the continuous spectrum (for |U| < 1) where there is no localized state, we construct the Green’s function using the expansion in terms of eigenfunctions of the continuous spectrum and obtain an expression for the time Green’s function in the form of a power series in U. It unexpectedly turns out that this series converges absolutely even in the case where the localized state is added to the continuous spectrum. We can therefore say that the Green’s function constructed using the states of the continuous spectrum also contains an implicit contribution from the localized state.
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. Finally, we present several short simulations of the scattering process, which emphasize how a simple methodology can be used to visualize some remarkable phenomena.