Physikalisches Wahlfach: Classical and Quantum Chaos (PW)5 ECTS
(englische Bezeichnung: Elective Course in Physics: Classical and Quantum Chaos)
(Prüfungsordnungsmodul: Weitere Module aus dem Wahlpflichtbereich I)

Lehrveranstaltungen:

• • Classical and Quantum Chaos
    (Vorlesung, 2 SWS, Sam Shallcross, Mo, 14:00 - 16:00, SR 00.732)
  • Classical and Quantum Chaos
    (Übung, 3 SWS, Sam Shallcross, Mi, 14:00 - 17:00, SR 01.332)

Inhalt:

Contents:
One of the most well known problems in classical mechanics is the Kepler problem: a two body problem with an inverse square force law that can be solved exactly. For many years it was assumed that although very few problems in classical mechanics can be solved exactly, the simplest cases such as the Kepler problem were generic: simple underlying equations should yield a simple dynamics, such as the stable planetary orbits of the solar system found in the Kepler problem. Beginning with the work of Poincare it came to be realized that this understanding of classical mechanics is fundamentally wrong: simple underlying equations (for instance 3 bodies interacting via a inverse square law) can generate highly complex non-periodic solutions. The closed orbits of the Kepler problem are in fact the exception in classical mechanics. The trajectories of a generic mechanical system are in fact chaotic: they are non-periodic and display an exponential sensitivity to initial conditions – the so called “butterfly effect” - such that the behaviour of the simple dynamical system becomes essentially unpredictable.
The idea that very simple systems can yield complex unpredictable behaviour lies at the heart of chaos theory. This course will study the behaviour of a number of the most well known systems for which chaos is found including the logistic map (used to study driven dissipative chaotic systems such as the driven dissipative pendulum or microbial populations) and the kicked oscillator (used to study conserving systems such as the solar system). Along the way we will learn that despite the seemingly vast disparity of chaotic systems there are many features that such systems share; this will be elucidated via renormalization group theory which we will examine closely in the context of the route to chaos of the logistic map.
Finally, time permitting, we will examine what happens when we consider the quantum versions of the various classical Hamiltonians that lead to chaotic behaviour. In quantum mechanics, of course, the dynamical trajectory is an illegitimate concept, momentum and position cannot be measured simultaneously, so how does chaos -- for which the concept of a trajectory would seem to be crucial -- manifest itself in quantum systems? These questions lead us to the subject of semi-classics: the study of systems for which Planck's constant is much smaller than the characteristic actions of the system or, in other words, the limit in which Planck's constant goes to zero. This limit is singular and as a consequence is much richer than the trivial regular limit of special relativity to yield Newtonian mechanics. We will consider the consequences of the unusual nature of the limit of classical mechanics out of quantum mechanics for the cases where the corresponding classical behaviour is chaotic.

Lernziele und Kompetenzen:

Learning goals and competences:
Students

• explain the relevant topics of the lecture

• apply the methods to specific examples

Verwendbarkeit des Moduls / Einpassung in den Musterstudienplan:

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

Studien-/Prüfungsleistungen:

Physikalisches Wahlfach: Classical and Quantum Chaos (Prüfungsnummer: 693143)

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

Anteil an der Berechnung der Modulnote: 100.0 %

Erstablegung: WS 2014/2015

1. Prüfer:

Sam Shallcross