Robert Alicki and Mark Fannes
- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter presents the basic mathematical facts concerning Schrödinger's equation with constant and time-dependent Hamiltonians: Stone's theorem on strongly continuous one-parameter groups of ...
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This chapter presents the basic mathematical facts concerning Schrödinger's equation with constant and time-dependent Hamiltonians: Stone's theorem on strongly continuous one-parameter groups of unitaries, the Kato–Rellich criterion for self-adjointness, and the Dyson expansion. In particular, Floquet's theory for periodic perturbations is outlined and illustrated by examples of kicked systems: the kicked top and the baker map. Then classical mechanics is introduced as a limit of quantum theory using coherent states and mean-field limits. The formalism of classical differentiable dynamics is briefly described and the classical and quantum aspects of the motion of a free particle on a compact Riemannian manifold are discussed including Weyl's theorem characterizing spectra of generalized Laplacians such as Beltrami–Laplace operators.Less
This chapter presents the basic mathematical facts concerning Schrödinger's equation with constant and time-dependent Hamiltonians: Stone's theorem on strongly continuous one-parameter groups of unitaries, the Kato–Rellich criterion for self-adjointness, and the Dyson expansion. In particular, Floquet's theory for periodic perturbations is outlined and illustrated by examples of kicked systems: the kicked top and the baker map. Then classical mechanics is introduced as a limit of quantum theory using coherent states and mean-field limits. The formalism of classical differentiable dynamics is briefly described and the classical and quantum aspects of the motion of a free particle on a compact Riemannian manifold are discussed including Weyl's theorem characterizing spectra of generalized Laplacians such as Beltrami–Laplace operators.
