01206nas a2200133   4500008004300000245007600043210006900119260001300188520077600201100002100977700001900998700001901017856003601036       2003                        en_Ud  00aEffective dynamics for Bloch electrons: Peierls substitution and beyond0 aEffective dynamics for Bloch electrons Peierls substitution and   bSpringer3 aWe consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, $\\\\phi(\\\\epsi x)$, and vector potential $A(\\\\epsi x)$, with $x \\\\in \\\\R^d$ and $\\\\epsi \\\\ll 1$. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of $L^2(\\\\R^d)$ and an effective Hamiltonian governing the evolution inside this subspace to all orders in $\\\\epsi$. To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics.1 aPanati, Gianluca1 aSpohn, Herbert1 aTeufel, Stefan  uhttp://hdl.handle.net/1963/3040