Fractional-Time Schrödinger Equation: Fractional Dynamics on a Comb

Details Fractional-Time Schrödinger Equation: Fractional Dynamics on a Comb Fractional-Time Schrödinger Equation: Fractional Dynamics on a Comb

2011-08-31

arxiv.org/abs/1108.6178v1

Chaos, Solitons & Fractals 44 (2011) 348-352

Physics

Quantum Physics

Scientific paper

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.

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