Past exam of the quantum condensed matter course of the University of Cambridge / 2009 / Section B / B1 / b

A wave packet in an energy band has group velocity $v={\hslash }^{-1}dE/dk$. The semiclassical force law $\hslash \dot{k}=F$ then gives $\dot{v}={\hslash }^{-2}(d^{2}E/d{k}^2)F$. The curvature therefore defines the one-dimensional effective mass by $1/m^{\ast }=E^{\prime \prime }/{\hslash }^2$. Here $E^{\prime \prime }(k)=A{a}^2\cos (ka)$, so at the band minimum $k=0$,It measures acceleration within the band, rather than a change in the electron's rest mass. At a band maximum the electron curvature mass can be negative; a missing electron is then described by a positively charged hole with a positive curvature mass.