TY - JOUR
T1 - Effective dynamics for Bloch electrons: Peierls substitution and beyond
Y1 - 2003
A1 - Gianluca Panati
A1 - Herbert Spohn
A1 - Stefan Teufel
AB - 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.
PB - Springer
UR - http://hdl.handle.net/1963/3040
U1 - 1293
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Space-adiabatic perturbation theory
JF - Adv. Theor. Math. Phys. 7 (2003) 145-204
Y1 - 2003
A1 - Gianluca Panati
A1 - Herbert Spohn
A1 - Stefan Teufel
AB - 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.
PB - International Press
UR - http://hdl.handle.net/1963/3041
U1 - 1292
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Space-adiabatic perturbation theory in quantum dynamics
JF - Physical review letters. 2002 Jun; 88(25 Pt 1):250405
Y1 - 2002
A1 - Gianluca Panati
A1 - Herbert Spohn
A1 - Stefan Teufel
AB - 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.
PB - American Physical Society
UR - http://hdl.handle.net/1963/5985
U1 - 5841
U2 - Mathematics
U3 - Mathematical Physics
U4 - -1
ER -