Semiclassical approximations for Hamiltonians with operator-valued symbols

TL;DR

This paper develops semiclassical approximations for quantum systems with operator-valued symbols, providing tools like Egorov's theorem and Wigner function transport to analyze the adiabatic and semiclassical limits, with applications to solid state physics.

Contribution

It introduces a rigorous mathematical framework for semiclassical analysis of Hamiltonians with operator-valued symbols, linking classical flows to quantum expectations and dynamics.

Findings

Derived a formula for equilibrium expectations in such systems

Proved an Egorov theorem for operator-valued symbols

Applied results to the Hofstadter model in solid state physics

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 $\varepsilon\ll 1$ controls the separation of time scales and the limit $\varepsilon\to 0$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $\varepsilon\to 0$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $\varepsilon$-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn, coming from an $\epsi$-dependent classical Hamilton function and an $\varepsilon$-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 $\varepsilon^2$. In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics. 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.