Semi-classical theory of matrix valued Hamiltonians (PW)5 ECTS
(englische Bezeichnung: Semi-classical theory of matrix valued Hamiltonians)
(Prüfungsordnungsmodul: Weitere Module aus dem Wahlpflichtbereich I)

Lectures:

• • Semi-classical theory of matrix valued Hamiltonians
    (Vorlesung, 2 SWS, Sam Shallcross, Fri, 13:00 - 15:00, SR 01.566)
  • Semi-classical theory of matrix valued Hamiltonians
    (Übung, 3 SWS, Sam Shallcross, Zeit und Raum n.V.)

Inhalt:

Contents:
The classical limit of quantum mechanics is singular; setting Planck's constant to zero in the Schrödinger equation does not recover any version of classical mechanics. Such singular limits bring with them emergent physics at the "borderlands" of the two theories - for example quantum chaos. In this lecture series we explore this borderland when the Hamiltonian of the quantum system is matrix valued, for example the Dirac-Weyl equation made famous by graphene, or the Dirac equation. The semi-classical theory for such Hamiltonians includes rich physics associated with geometric phases and Berry curvatures, and we will explore these concepts via a series of concrete examples in solid state physics.

Subjects covered:
The Berry phase, Berry curvature, Holonomy as a concept in physics, examples of the Berry phase including the Aharonov-Bohn effect, basic semi-classical theory e.g. WKB theory, the Gutzwiller trace formula, Maslov phases, examples including graphene, bilayer graphene, the graphene twist bilayer, silicene, topological insulators such as SnTe.

Lernziele und Kompetenzen:

Learning goals and competences:
Students

• explain the relevant topics of the lecture

• apply the methods to specific examples

Literatur:

Literature
Introductory treatment of semi-classics:
[1] L. D. Landau and E. M. Lifshitz. Vol. 3. Quantum mechanics; non-relativistic theory (3ed., Pergamon, 1991) - a classic work that has a good chapter on semi-classical theory, treats only effectively one-dimensional problems i.e. WKB theory.
Geometric phases in physics:
[2] A. Shapere and F. Wilczek, Geometric phases in physics (WS, 1989) (ISBN 9971505991) - a selection of articles that contains many of the classic papers on the subject.
[3] D. Chruscinski and A. Jamiolkowski, Geometric phases in classical and quantum mechanics (Birkhauser, 2004) (ISBN 081764282X) - rather mathematical.
Quantum chaos and semi-classics:
[4] chaosbook.org - excellent on-line resource by one of the pioneers in the field. Treats quantum chaos in depth but also fully describes modern semi-classical theory.
[5] Martin Gutzwiller, Chaos in classical and quantum mechanics (Springer 1990) - an intuitive discussion of semi-classics with a focus on quantum chaos in particular atoms in magnetic fields.
Graphene
[6] Pati S.K., Enoki T., Rao C.N.R. Graphene and Its Fascinating Attributes (WS, 2011) (ISBN 9814329355) - a selection of articles on graphene.

Verwendbarkeit des Moduls / Einpassung in den Musterstudienplan:

1. Physik (1. Staatsprüfung für das Lehramt an Gymnasien)
   (Po-Vers. 2010 | NatFak | Physik (1. Staatsprüfung für das Lehramt an Gymnasien) | Module Fachwissenschaft Physik | Wahlpflichtbereich | Weitere Module aus dem Wahlpflichtbereich I)

Studien-/Prüfungsleistungen:

Semi-classical theory of matrix valued Hamiltonians (Prüfungsnummer: 821637)

(englischer Titel: Semi-classical theory of matrix valued Hamiltonians)

Prüfungsleistung, mündliche Prüfung, Dauer (in Minuten): 30, benotet, 5 ECTS

Anteil an der Berechnung der Modulnote: 100.0 %

Erstablegung: SS 2016, 1. Wdh.: SS 2016 (nur für Wiederholer)

1. Prüfer:

Sam Shallcross