Stefan Teufel

Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond

AbstractWe consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, φ(ɛx), and vector potential A(ɛx), with xℝd and ɛ≪1. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of L2(ℝd) and an effective Hamiltonian governing the evolution inside this subspace to all orders in ɛ. 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.

Space-adiabatic perturbation theory

3.1 Almost invariant subspaces 3.2 Mapping to the reference space 3.3 Effective dynamics 3.3.1 Expanding the effective Hamiltonian 3.4 Semiclassical limit for effective Hamiltonians 3.4.1 Semiclassical analysis for matrix-valued symbols 3.4.2 Geometrical interpretation: the generalized Berry connection 3.4.3 Semiclassical observables and an Egorov theorem

Adiabatic Decoupling and Time-Dependent Born-Oppenheimer Theory

Abstract: We reconsider the time-dependent Born–Oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands. Band crossings are allowed and our results are local in the sense that they hold up to the first time when a band crossing is encountered. The adiabatic decoupling leads to an effective Schrödinger equation for the nuclei, including contributions from the Berry connection.

Quantum Spacetime without Observers: Ontological Clarity and the Conceptual Foundations of Quantum Gravity

The term “3-geometry” makes sense as well in quantum geometrodynamics as in classical theory. So does superspace. But space-time does not. Give a 3-geometry, and give its time rate of change.

The time-dependent Born-Oppenheimer approximation

We explain why the conventional argument for deriving the time-dependent Born-Oppenheimer approximation is incomplete and review recent mathematical results, which clarify the situation and at the same time provide a systematic scheme for higher order corrections. We also present a new elementary derivation of the correct second-order time-dependent Born-Oppenheimer approximation and discuss as applications the dynamics near a conical intersection of potential surfaces and reactive scattering.

Simple Proof for Global Existence of Bohmian Trajectories

We address the question whether Bohmian trajectories exist for all times. Bohmian trajectories are solutions of an ordinary differential equation involving a wavefunction obeying either the Schrödinger or the Dirac equation. Some trajectories may end in finite time, for example by running into a node of the wavefunction, where the law of motion is ill-defined. The aim is to show, under suitable assumptions on the initial wavefunction and the potential, global existence of almost all solutions. We provide an alternative proof of the known global existence result for spinless Schrödinger particles and extend the result to particles with spin, to the presence of magnetic fields, and to Dirac wavefunctions. Our main new result is conditions on the current vector field on configuration-space-time which are sufficient for almost-sure global existence.

Semiclassical Limit for the Schrödinger Equation¶with a Short Scale Periodic Potential

Abstract: We consider the dynamics generated by the Schrödinger operator H=−½Δ+V(x)+W(ɛx), where V is a lattice periodic potential and W an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit ɛ→ 0 the time dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined by the semiclassical dynamics.

A Note on the Adiabatic Theorem Without Gap Condition

We simplify the proof of the adiabatic theorem of quantum mechanics without gap condition of Avron and Elgart by providing an elementary solution of the ‘commutator equation’. In addition, a minor modification of their argument allows for more direct treatment of eigenvalue crossings. We also obtain simple, explicit conditions that yield information on the rate of convergence in the adiabatic limit.

Effective Hamiltonians for Constrained Quantum Systems

Introduction Main results Proof of the main results The whole story Appendix A. Geometric definitions and conventions Bibliography

The flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs

The quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. The relation between these crossing probabilities and the usual formula for the scattering cross section is provided by the flux-across-surfaces theorem, which was conjectured by Combes, Newton, and Shtokhamer [Phys. Rev. D 11, 366–372 (1975)]. We prove the flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs. The proof is based on the free flux-across-surfaces theorem (Daumer et al.) [Lett. Math. Phys. 38, 103–116 (1996)], and on smoothness properties of generalized eigenfunctions: It is shown that if the potential V(x) decays like |x|−γ at infinity with γ>n∈N then the generalized eigenfunctions of the corresponding Hamiltonian −1/2Δ+V are n−2 times continuously differentiable with respect to the momentum variable.