Long wavelength approximation of transport processes in a single-band crystal

The single band, long wavelength approximation (SBA-LWA) is currently used in textbooks as a quasi-free-particle picture of the motion in a quantum crystal. The resulting transport process might thereby look a trivial issue. In contrast, we shall show that the SBA-LWA hides some controversial aspects that should be clarified at the level of an advanced course of condensed matter physics, and refer to the incompleteness of the SBA representation. In particular, it will be shown that the single-band velocity v(1B), expressed in terms of the projectors on the Bloch states, cannot be a transport velocity in a full sense, since the resulting current violates the continuity equation. The drawback manifests itself as a 'lost' current J(lost), which provides a non conventional estimate of the limits of accuracy of SBA-LWA. The vanishing of Jlost corresponds to the effective mass approximation in which the dispersion relation can be reduced to a quadratic form in the (pseudo) momentum components. In practice, the quantity transported by v(1B) is not the bare mass, but the effective mass, until this notion does make sense. Recalling that the non-quadratic expression of the relativistic kinetic energy leads to a difference between the rest and moving mass, the notion of the lost current is finally used as a non-conventional approach to relativistic quantum mechanics, with special reference to Dirac's theory.