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Quantum Reports

Wave Packet Propagation in Anisotropic Media: Interplay of Comb Geometry and a Dirac δ-Barrier

Ognen Kapetanoski, Ervin Kaminski Lenzi, Irina Petreska

Published Sep 22, 2026• 0 citations

Abstract

We investigate numerically the time-dependent Schrödinger equation on a two-dimensional comb in the presence of a localized Dirac delta potential on the backbone. The geometric constraint and the point interaction are both regularized via narrow Gaussian approximations, and the resulting equation is discretized with second-order finite differences and integrated with a fourth-order Runge–Kutta scheme on a finite domain. Two initial states are considered—a Gaussian wave packet and a comb-like wave function—and two barrier configurations are studied: a potential centered at the origin and one displaced along the backbone. In both cases, the Dirac delta interaction suppresses the probability density at the barrier position and reorganizes the wave packet into a persistent split-peak profile, in contrast to the single spreading maximum observed in the free case. When the barrier is displaced, due to reflection from the barrier, the left–right symmetry is broken and the density remains asymmetrically concentrated near the interaction site. To elucidate the results further, we also analyze the second moment, which undoubtedly confirms that the anomalous dynamics features, such as the subdiffusive behavior on comb-like structures in classical systems are retained in the quantum case as well.

DOI: 10.3390/quantum8040099

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Localization of relativistic particles have been of great research interests over many decades. We investigate the time evolution of the Gaussian wave packets governed by the one dimensional Dirac equation. For the free Dirac equation, we obtain the evolution profiles analytically in many approximation regimes, and numerical simulations consistent with other numerical schemes. Interesting behaviors such as Zitterbewegung and Klein paradox are exhibited. In particular, the dispersion rate as a function of mass is calculated, and it yields an interesting result that super-massive and massless particles both exhibit no dispersion in free space. For the Dirac equation with random potential or mass, we employ the Chebyshev polynomials expansion of the propagator operator to numerically investigate the probability profiles of the displacement distribution when the potential or mass is uniformly distributed. We observe that the widths of the Gaussian wave packets decrease approximately with the power law of order

o(s^{-ν})

with $\frac{1}{2}

In this work, we have investigated the Gaussian and Airy wave packets colliding on the Dirac delta potentials. We have used the time-dependent space-fractional Schrodinger equation and solved it by using the slit step finite difference method. Here, we have presented some exclusive exotic propagation properties of Dirac delta potential that cannot be observed for other very similar potentials such as one-dimensional coulomb potential. We can now rectify any number of colliding wave packets provided that the correct parameters adjustment is possible. We have also addressed the wave packet speed control, wave packet splitting, wave packet rectification through collision process, wave packet dispersion, etc. The effects of different parameters such as the nonlinearity, fractitionality, dispersion, etc. on the wave packet-delta potential collision are also considered.

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