Anomalous quantum mechanics

TL;DR

This paper introduces fractional operators and equations in quantum mechanics to model anomalous behaviors in complex media, leading to new predictions like an anomalous hydrogen atom with higher transition energies.

Contribution

It develops fractional Schrödinger equations and virial theorem, providing a novel framework for anomalous quantum phenomena in heterogeneous media.

Findings

Derived fractional Schrödinger equations for free particles and potentials.

Established an anomalous hydrogen atom with higher transition energies.

Discussed the equivalence of fractional and anomalous Heisenberg pictures.

Abstract

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 Schr\"odinger 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 Schr\"odinger picture is also discussed.