*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/9780198504009.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Quantum systems with a discrete dynamical spectrum, such as finite dimensional systems or particles confined to a finite volume, cannot exhibit true ergodic behaviour. Nevertheless, scaling limits ...
More

Quantum systems with a discrete dynamical spectrum, such as finite dimensional systems or particles confined to a finite volume, cannot exhibit true ergodic behaviour. Nevertheless, scaling limits can be considered. This chapter focuses on the behaviour of entropy for small value of Planck's constant, i.e. how the classical limit is reached. It also considers other classical limits such as the limit of large angular momentum for the kicked top. Numerical computations are needed to compute the actual value of the total entropy produced up to a finite time, but its saturation behaviour for long times can be established analytically. The operational partitions that eventually lead to the largest entropy can also be determined. As a complementary tool for the analysis of randomizing behaviour, the chapter sketches the use of the Gram matrix to study the statistics of return times.Less

Quantum systems with a discrete dynamical spectrum, such as finite dimensional systems or particles confined to a finite volume, cannot exhibit true ergodic behaviour. Nevertheless, scaling limits can be considered. This chapter focuses on the behaviour of entropy for small value of Planck's constant, i.e. how the classical limit is reached. It also considers other classical limits such as the limit of large angular momentum for the kicked top. Numerical computations are needed to compute the actual value of the total entropy produced up to a finite time, but its saturation behaviour for long times can be established analytically. The operational partitions that eventually lead to the largest entropy can also be determined. As a complementary tool for the analysis of randomizing behaviour, the chapter sketches the use of the Gram matrix to study the statistics of return times.

*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 ...
More

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.