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Semiclassical Approximations for Hamiltonians with Operator-Valued Symbols.
- Source :
- Communications in Mathematical Physics; Jun2013, Vol. 320 Issue 3, p821-849, 29p
- Publication Year :
- 2013
-
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 (Phys Rev A (3) 44(8):5239-5256, ), coming from an $${\varepsilon}$$ -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 (Xiao et al. in Rev Mod Phys 82(3):1959-2007, ). 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00103616
- Volume :
- 320
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Communications in Mathematical Physics
- Publication Type :
- Academic Journal
- Accession number :
- 87784612
- Full Text :
- https://doi.org/10.1007/s00220-012-1650-5