Nonadiabatic effects in the nuclear probability and flux densities through the fractional Schrödinger equation

Abstract

Nonadiabatic effects in the nuclear dynamics of the H2+ molecular ion, initiated by ionization of the H2 molecule, is studied by means of the probability and flux distribution functions arising from the space fractional Schrödinger equation. In order to solve the fractional Schrödinger eigenvalue equation, it is shown that the quantum Riesz fractional derivative operator fulfills the usual properties of the quantum momentum operator acting on the bra and ket vectors of the abstract Hilbert space. Then, the fractional Fourier grid Hamiltonian method is implemented and applied to molecular vibrations. The eigenenergies and eigenfunctions of the fractional Schrödinger equation describing the vibrational motion of the H2+ and D2+ molecules are analyzed. In particular, it is shown that the position‐momentum Heisenberg's uncertainty relationship holds independently of the fractional Schrödinger equation. Finally, the probability and flux distributions are presented, demonstrating the applicability of the fractional Schrödinger equation for taking into account nonadiabatic effects.

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

“…However, its application to quantum mechanics was started by the pioneering work of the Laskin and others . Based on this outstanding idea, intense investigations have been carried out in different fields of studies, for example, energy band structure for the periodic potential, position‐dependent mass fractional Schrödinger equation, fractional quantum oscillator, nuclear dynamics of the H 2 + molecular ion, propagation dynamics of a light beam, spatial soliton propagation, solitons in the fractional Schrödinger equation with parity‐time‐symmetric lattice potential, gap solitons, Rabi oscillations in a fractional Schrödinger equation, self‐focusing, and wave collapse, elliptic solitons, light propagation in honeycomb lattice, and so on. These studies are based on the different methods such as domain decomposition method, energy conservative difference scheme, conservative finite element method, fractional Fan subequation method, split‐step Fourier spectral method, transfer‐matrix method, and so on.…”

Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation

“…Initially, the FSE was introduced as a quantum-mechanical model to investigate particle motion governed by Lévy flights, employing the Feynman-integral formalism [34]. Although almost all contributions to this have been developed along with the traditional Schrödinger equation, the FSE exhibits intriguing quantum phenomena due to its fractional derivative index k [35][36][37][38][39][40][41][42][43][44][45][46]. These investigations cover a wide range of topics, including energy band structures, light beam propagation dynamics, position-dependent mass FSE, the nuclear dynamics of molecular ion H 2 + , Rabi oscillations, spatial soliton propagation, fractional harmonic oscillators, and others.…”

Quantum tunneling from family of Cantor potentials in fractional quantum mechanics

“…The domain of SFQM have grown fast over the last two decades and various applications are discussed by different authors. Some of the notable work are the energy band structure for the periodic potential [6], position-dependent mass fractional Schrodinger equation [7], fractional quantum oscillator [8], nuclear dynamics of the H + 2 molecular ion [9], propagation dynamics of a light beam [10], spatial soliton propagation [11], solitons in the fractional Schrodinger equation with parity-time-symmetric lattice potential [12], gap solitons [13], Rabi oscillations in a fractional Schrodinger equation [14], self-focusing and wave collapse [15], elliptic solitons [16], light propagation in a honeycomb lattice [17], scattering features in non-Hermitian SFQM [18], tunneling time [19,20] etc. Different methods are used in such studies such as domain decomposition method [21], energy conservative difference scheme [22], conservative finite element method [23], fractional Fan sub-equation method [24], split-step Fourier spectral method [25], transfer-matrix method [26] etc.…”