Fractional-Time Schr\"odinger Equation: Fractional Dynamics on a Comb

TL;DR

This paper critically examines the fractional-time Schr"odinger equation, demonstrating through a quantum comb model that naive fractional derivatives can lead to loss of quantum information and are insufficient for accurate quantum dynamics modeling.

Contribution

It introduces a quantum comb model showing that fractional derivatives in the Schr"odinger equation can be inadequate for describing quantum processes.

Findings

FTSE is a special case of the quantum comb model at α=1/2

Naive fractional derivatives cause loss of quantum information

FTSE may not accurately describe quantum dynamics

Abstract

The physical relevance of the fractional time derivative in quantum mechanics is discussed. It is shown that the introduction of the fractional time Scr\"odinger equation (FTSE) in quantum mechanics by analogy with the fractional diffusion $\frac{\prt}{\prt t}\rightarrow \frac{\prt^{\alpha}}{\prt t^{\alpha}}$ can lead to an essential deficiency in the quantum mechanical description, and needs special care. To shed light on this situation, a quantum comb model is introduced. It is shown that for $\alpha=1/2$, the FTSE is a particular case of the quantum comb model. This \textit{exact} example shows that the FTSE is insufficient to describe a quantum process, and the appearance of the fractional time derivative by a simple change $\frac{\prt}{\prt t}\rightarrow \frac{\prt^{\alpha}}{\prt t^{\alpha}}$ in the Schr\"odinger equation leads to the loss of most of the information about quantum dynamics.