Comment on "The quantum mechanics of electric conduction in crystals," by R. J. Olsen and G. Vignale [Am. J. Phys. 78 (9), 954-960 (2010)]

TL;DR

This paper clarifies the physical meaning of scattering amplitude transformations using time-delay concepts and revisits Hartman's effect, demonstrating that group velocity can become arbitrarily large in an infinite periodic potential chain.

Contribution

It provides a physical interpretation of scattering amplitude transformations and extends the analysis of Hartman's effect to infinite periodic potentials.

Findings

Transformation properties relate to potential displacement via time-delay.

Group velocity can become arbitrarily large in infinite periodic potentials.

Revised understanding of Hartman's effect in periodic structures.

Abstract

In this note we use the notion of time-delay to explain the physical content of the transformation properties of transmission and reflection amplitudes, as a result of a displacement of the potential. Then, we reconsider the recent analysis of the scattering problem by a finite-periodic potential, by Olsen and Vignale, to obtain the total reflection condition in the limit of an infinite number of cells. In doing this, we obtain an expression of Hartman's effect, showing that the group velocity of the transmitted particle inside the potential chain can become arbitrary large, as the number of cells tends to infinity.