Fractional Heisenberg equation

Aniello Lampo, Miguel Ángel García March, Maciej Lewenstein

Alexander Iomin

Fractional time (non-unitary) quantum mechanics is discussed for both Schrödinger and Heisenberg representations of quantum mechanics. A correct form of the fractional Schrödinger equation is elucidated. A generic regularization procedure for the fractional Heisenberg equations of motion is suggested that makes it possible to solve these nonlinear equations in the framework of photonic and spin coherent states consideration.

Leidy Yoana Medina, Francisco Núñez‐Zarur, Jhon Fredy Pérez-Torres

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.

Shue-Kwing Chan, Kiang-Chuen Young

S. Yu. Gubanov

Hong Zhang

The fractional operators together with exponential quantum in coordinate and momentum space corresponding to the power of observables are introduced. Based on an exponential relation between energy and momentum, the fractional Schrödinger equations for the free particle and the one in potential fields in heterogeneous complex media are found. The fractional equation of motion and the fractional virial theorem for anomalous quantum mechanics are then developed. Applying the fractional virial theorem, we derive an anomalous hydrogen atom whose transition energy values are much higher than that of Bohr hydrogen atom. The anomalous Heisenberg picture being equivalent to the fractional Schrödinger picture is also discussed.