Undergraduate research projects 2015-16
Sebastian Müller
1. Classical perturbation theory in astronomy (level 6, 10-20cp). Many systems in nature are
non-integrable. Roughly speaking this means that there are not enough conserved quantities to
find an analytical solution for the equations of motion. However for systems that are close to
being integrable it is possible to find approximate solutions. The technique used to get these
approximate solutions (classical perturbation theory) is based on Hamiltonian mechanics.
Classical perturbation theory has many applications in astronomy. For example the shape of the
rings of Saturn can be obtained using perturbation theory and some results on integrable
systems. In this project one of these problems should be studied in detail, along with the
techniques of perturbation theory. The project will involve reading and analytical and/or numerical
work. Prerequisite: Mechanics 2/23 or equivalent. References: Parts of S. Wimberger, Nonlinear
Dynamics and Quantum Chaos: An Introduction; M. Berry, Regular and irregular motion,
https://michaelberryphysics.files.wordpress.com/2013/07/berry076.pdf
2. Solar cells (Level 6 or 7, 20+cp). Solar cells are composed of two materials (semiconductors).
When light shines on the cell it raises electrons in these materials to excited levels. The excited
electrons have a tendency to move from one material to the other and this leads to a voltage
difference that can be used as a source of energy. The aim of this project is to solve differential
equations modelling the time evolution of relevant variables such as the concentration of excited
electrons in the two materials, for some simple geometries. As this project is quite research
oriented there is scope for an extended version at level 7. Prerequisites: Applied Partial
Differential Equations or Ordinary Differential Equations, preferably both. For a level 7 project
along these lines further units related to differential equations or Statistical Mechanics may be
helpful. Reference: Parts of 'The Physics of Solar Cells' by Jenny Nelson
3. Quantum gravity and Hamiltonian mechanics (level 6 or 7, 20-40cp). One of the fundamental
problems of Mathematical Physics is to find a quantum mechanical description of gravity. In one
approach to this problem (loop quantum gravity) space itself becomes quantised, and the volume
of a part of space becomes an operator similar to the Hamiltonian operator in quantum
mechanics. There is also an associated classical Hamiltonian in the sense of Hamiltonian
mechanics. This gives the volume of a part of space of space for example of the shape of a
tetrahedron or a pentahedron. The Hamiltonian associated to the tetrahedron is integrable and
can be treated using the method of action and angle variables introduced in the end of the
Mechanics 2/23 course. Very recent research indicates that the the Hamiltonian associated to the
pentahedron is not integrable. In this project students will study this nonintegrable Hamiltonian
and for example (i) perform numerical experiments (so called Poincare plots) that help to
determine for which parameter values the system is to close to being integrable; (ii) apply
analytical approximation techniques that allow to describe near integrable systems (canonical
perturbation theory); (iii) study elements of the quantum mechanical background. The precise
selection of topics will depend on the level and size of the project, and on the interests of the
student. A level 6 project along these lines could essentially be a Mechanics project.
Prerequisites: Mechanics 2/23. Quantum Mechanics (or its equivalent in Physics) is a
prerequisite for a level 7 project and may be useful to take in parallel to a level 6 project.
References: Depend heavily on the focus of the project, one reference will be C.E. ColemanSmith and B. Müller, A ''Helium atom'' of space: Dynamical instability of the isochoric
pentahedron, Phys. Rev. D 87, 044047 (2013) http://arxiv.org/abs/1212.1930.
4. Quantum chaos and the local density of states (level 7, 20-40cp) In quantum chaos one
considers systems whose classical motion is chaotic, i.e., depends sensitively on the initial
conditions. One then investigates the quantum mechanical properties of these systems, for
example their energy levels. It turns out that there are deep connections between the classical
and quantum mechanical behaviour. For example the level density (a sum over delta functions
Annals located at the energy levels) of a chaotic quantum system can be approximated by an
expression that involves (periodic) classical trajectories. This expression can be used as a
starting point to determine statistical properties of the distribution of energy levels. One then sees
that these statistical properties are universal for all chaotic systems and that the energy levels
always seem to 'repel' each other. The aim of this project is to generalise key ingredients of
the underlying theory to the local density of states. This differs from the density considered
previously because the delta function corresponding to each energy level is multiplied with the
squared absolute value of the corresponding energy eigenfunction at a given point. Related
courses: Quantum Mechanics (or its equivalent in Physics) is a prerequisite. Mechanics 2/23
would be helpful. It is advisable to take Quantum Chaos in parallel. Reference: Parts of 'Quantum
Chaos: An Introduction' by Hans-Juergen Stöckmann; ‘Nonlinear Dynamics and Quantum Chaos:
An Introduction’ by Sandro Wimberger.
5. Random matrix theory for many-particle systems (level 7, 20-40cp). Beyond the ensembles
of matrices introduced in the Random Matrix Theory course there are additional ensembles that
can be used to model quantum mechanical systems about which one has more information than
just whether the Hamiltonian is real symmetric or general hermitian. Such ensembles have been
introduced for example to study systems with several particles. In this case the quantum
mechanical state of the full system is determined by the states of the individual particles making
up the system. In many situations there are additional conditions that the Hamiltonian of a many
particle system has to satisfy, e.g., it may be a sum of operators that each modify the state of only
one or two particles making up the system. This project involves studying parts of the literature
about so-called embedded ensembles for many-particle systems, and research e.g. along one of
the following lines: numerical investigations; studying systems that whose Hamiltonian involves a
combination of one- and two-particles terms by combining results about these two types of
operators; investigating examples where few particles are present in the system. Related
courses: Random Matrix Theory and Quantum Mechanics are prerequisites, the part of the
Statistical Mechanics course about quantum mechanics for many-particle systems is helpful but
not a prerequisite. Reference: e.g. R. Small and S. Müller, Particle Diagrams and Statistics of
Many-Body Random Potentials, http://arxiv.org/abs/1412.2952
6. Projects about other aspects of Quantum Chaos and Mathematical Physics research, and Group
Projects. Topics to be decided in conversation with interested students.