%0 Journal Article
%D 2003
%T Effective dynamics for Bloch electrons: Peierls substitution and beyond
%A Gianluca Panati
%A Herbert Spohn
%A Stefan Teufel
%X We 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.
%I Springer
%G en_US
%U http://hdl.handle.net/1963/3040
%1 1293
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-08T12:11:51Z\\nNo. of bitstreams: 1\\n0212041v2.pdf: 395908 bytes, checksum: 7b198d3311402d34945989b8f6edd8b8 (MD5)
%0 Journal Article
%J Adv. Theor. Math. Phys. 7 (2003) 145-204
%D 2003
%T Space-adiabatic perturbation theory
%A Gianluca Panati
%A Herbert Spohn
%A Stefan Teufel
%X We study approximate solutions to the Schr\\\\\\\"odinger equation $i\\\\epsi\\\\partial\\\\psi_t(x)/\\\\partial t = H(x,-i\\\\epsi\\\\nabla_x) \\\\psi_t(x)$ with the Hamiltonian given as the Weyl quantization of the symbol $H(q,p)$ taking values in the space of bounded operators on the Hilbert space $\\\\Hi_{\\\\rm f}$ of fast ``internal\\\'\\\' degrees of freedom. By assumption $H(q,p)$ has an isolated energy band. Using a method of Nenciu and Sordoni \\\\cite{NS} we prove that interband transitions are suppressed to any order in $\\\\epsi$. As a consequence, associated to that energy band there exists a subspace of $L^2(\\\\mathbb{R}^d,\\\\Hi _{\\\\rm f})$ almost invariant under the unitary time evolution. We develop a systematic perturbation scheme for the computation of effective Hamiltonians which govern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the time-adiabatic theory.
%B Adv. Theor. Math. Phys. 7 (2003) 145-204
%I International Press
%G en_US
%U http://hdl.handle.net/1963/3041
%1 1292
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-08T12:17:36Z\\nNo. of bitstreams: 1\\n0201055v3.pdf: 449361 bytes, checksum: a37ea04fc4a4f59a75d03e4b2ec3df16 (MD5)
%0 Journal Article
%J Physical review letters. 2002 Jun; 88(25 Pt 1):250405
%D 2002
%T Space-adiabatic perturbation theory in quantum dynamics
%A Gianluca Panati
%A Herbert Spohn
%A Stefan Teufel
%X A systematic perturbation scheme is developed for approximate solutions to the time-dependent SchrÃ¶dinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band subspaces are suppressed to all orders, and the dynamics operates in that subspace in terms of an effective intraband Hamiltonian. As novel applications, we discuss the Born-Oppenheimer theory to second order and derive for the first time the nonperturbative definition of the g factor of the electron within nonrelativistic quantum electrodynamics.
%B Physical review letters. 2002 Jun; 88(25 Pt 1):250405
%I American Physical Society
%G en
%U http://hdl.handle.net/1963/5985
%1 5841
%2 Mathematics
%3 Mathematical Physics
%4 -1
%$ Submitted by Marta Maurutto (maurutto@sissa.it) on 2012-07-15T14:04:06Z\\nNo. of bitstreams: 0
%R 10.1103/PhysRevLett.88.250405