Semiclassical Approximations for Hamiltonians with Operator-Valued Symbols

Abstract

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter ε ≪ 1 controls the separation of time scales and the limit ε → 0 corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time ε → 0 is the semiclassical limit for the slow degrees of freedom. In this paper we show that the ε-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn [LF91], coming from an ε-dependent classical Hamilton function and an ε-dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order ε 2 . In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics [XCN10]. Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation.

Canonical Quantum Observables for Molecular Systems Approximated by Ab Initio Molecular Dynamics

“…Although this result is not new, see [15], we include a proof since it is important for the work here.…”

“…Tr A(y, p)B(y, p) dp dy , using the change of variables (y, y ′ ) = (x + x ′ )/2, x − x ′ , which verifies the claim. 6 The isometry between Weyl operators with the Hilbert-Schmidt inner product, Tr ( * B ), and the corresponding L 2 (R N × R N , C d×d ) symbols obtained by Lemma 2.1 shows how to extend from symbols in S to L 2 (R N × R N , C d×d ) by density of S in L 2 (R N × R N , C d×d ), see [15]. We will use the Hilbert-Schmidt norm  2 HS = Tr ( *  ) = Tr ( 2 ), to estimate Weyl operators.…”

Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature

“…The ǫ ↓ 0 limit of (1.1) has been studied by other methods. For example, by space-adiabatic perturbation theory [30][31][39] [38], and by studying the propagation of Wigner functions associated to the solution of (1.1) [26][2] [6]. The Wigner function approach is notable in that it has been used to study the propagation of wavepacket solutions of (1.1) through band crossings [25] [15].…”

Derivation of Ray Optics Equations in Photonic Crystals via a Semiclassical Limit

“…The reason the reality of electromagnetic fields complicates matters is that real electromagnetic fields are necessarily a linear combination of states associated to N positive and N negative frequency bands, i. e. at least two. While there are multiband semiclassical techniques available [BR90; LF91], we rely on a result by Teufel and Stiepan [ST13] which works only for single bands. Because of the reality of electromagnetic fields, we first use symmetry arguments to reduce everything to the positive frequency bands (cf.…”