In this study, we present a numerical investigation of wave packet dynamics in a nonlinear and dispersive medium described by the space-fractional Schrödinger equation, with direct relevance to quantum electronic device applications. Employing the Split-Step Finite Difference (SSFD) method, we analyse the superarrival phenomenon, where the arrival time of a Gaussian wave packet is accelerated due to the presence of a decelerating potential barrier. Two key configurations are explored: a barrier approaching the wave packet and a receding one. We show that the superarrival response is highly sensitive to system parameters such as the fractional order, nonlinearity, dispersion, and the barrier's motion profile. Our findings demonstrate that superarrival can be effectively tuned, offering new design strategies for emerging quantum electronic components such as ultrafast signal switches, wave-based logic gates, and controllable tunneling junctions in nano-engineered systems. This work bridges fundamental quantum transport with the functionality of next-generation electronic devices.
Superarrival, a quantum effect where a wave packet arrives earlier than its free-space counterpart, is investigated in fractional nanostructured media. We model a Gaussian wave packet interacting with a uniformly decelerating potential barrier or well using the fractional Schrödinger equation and the Split-Step Finite Difference method. The impact of the Lévy stability index, wave packet width, and initial distance on superarrival is analyzed. Results show that fractional media strongly enhance superarrival, especially for wider, low-energy packets interacting with barriers. These findings underscore the role of nonlocality in quantum transport and suggest potential applications in nanoscale optoelectronic devices such as tunable delay lines, ultrafast modulators, and quantum information processors.
We investigate a traveling Gaussian wave packet transport through a rectangular quantum barrier of lévy crystals in fractional quantum mechanics formalism. We study both standard and fractional Schrödinger equations in linear and nonlinear regimes by using a split-step finite difference (SSFD) method. We evaluate the reflection, trapping, and transmission coefficients of the wave packet and the wave packet spreading by using time-dependent inverse participation ratio (IPR) and second moment. By simultaneously adjusting the fractional and nonlinear terms, we create sharp pulses, which is an essential issue in optoelectronic devices. We illustrate that the effects of barrier height and width on the transmission coefficient are strangely different for the standard and fractional Schrödinger equations. We observe fortunately soliton-like localized wave packets in the fractional regime. Thus, we can effectively control the behavior of the wave evolution by adjusting the available parameters, which can excite new ideas in optics.
This paper presents a robust computational study of the Time-Dependent Schrödinger Equation (TDSE) dynamics for a spatial Gaussian wave packet confined within a one-dimensional infinite potential well. Employing a discrete Finite Difference Method (FDM) framework, we construct a tridiagonal Hamiltonian matrix to compute the unitary real-time evolution of the system. We systematically analyze two critical quantum phenomena: (1) scattering and quantum tunneling against an engineered central rectangular potential barrier, and (2) temporal dispersion and subsequent fractional quantum revival of the localized state. The numerical solver successfully resolved a distinct fractional revival and constructive self-interference peak at t = 1.416 time units. These computational benchmarks provide a rigorous pedagogical and diagnostic baseline for grid-based matrix methods in low-dimensional quantum mechanics.
Fractional calculus has become an essential framework in geophysics, optics, and biological systems to capture long-range correlations and anomalous transport. In this article, we extend this framework to explore a particle in a periodic potential, where the Schrodinger equation is extended to its fractional form. This allows us to study how the Levy index
q governs the formation and inversion of energy bands, offering a pathway to engineer new physical behaviors by tuning
q in periodic quantum systems. We solve the fractional Schrodinger equation (FSE) for periodic rectangular potentials of varying height
V0, barrier thickness
L, and well width
W using an imaginary-time evolution algorithm, supplemented by Gaussian process regression. This analysis reveals a qualitative shift in the system's band structure at
q=2, separating into distinct regimes of dispersion for
q>2 and
q<2. For
q>2, the energy bands invert as symmetric minima emerge within the first Brillouin zone and shift from
k=0 toward
k=±π/a with increasing
q. These degenerate minima define a Bloch-momentum qubit, suggesting an analog to valley degrees of freedom in valleytronics. The
q at which inversion completes scales as
q∝V0−0.28±0.05,
q∝L−0.35±0.08, and
q∝W−0.49±0.06 when varying parameters individually, indicating a tunable sensitivity to potential geometry. In contrast, for
q<2, the ground band hardens around
k=0, with a dispersion following
C∣k∣q+E0 near
k=0. This suggests an effective mass of 0 for
1<q<2 at the band's lowest energy state. These results demonstrate that the Levy index serves as a tunable degree of freedom in quantum periodic systems, capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics through effective-mass control.
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.
Scattering of a Gaussian wavepacket from rectangular potential barriers with increasing widths or heights is studied numerically. It is seen that during a certain time interval the time-evolving transmission probability increases compared to the corresponding unperturbed cases. In the literature this effect is known as superarrival in transmission probability. We present a trajectory-based explanation for this effect by using the concept of quantum potential energy and computing a selection of Bohmian trajectories. Relevant parameters in superarrivals are determined for the case that the barrier width increases linearly during the dispersion of the wavepacket. Nonlinearity in time perturbation is also considered.