The learning objectives are summarized below:
Emphasis in the learning objectives is on the physical interpretation of the results obtained with the mathematical and/or numerical models
- Master a specific subject from a short introduction and literature, and actively working on specific problems on that subject.
- Train yourself in the application of new mathematical and computational techniques for physics.
- Translation of a physical system to a mathematical description or a numerical model.
- Interpretation of the results from the mathematical and numerical models in terms of physics.
- Develop a healthy critical attitude towards written scientific material. Recognize the main thread in a scientific paper.
- Explore new area’s and make a comprehensive explanation for fellow students.
The course consists of three parts; all students take the first part, and then choose between either a mathematical or a numerical track. These tracks consist of two parts, each lasting approximately 2 weeks. There is a possibility to switch between the two tracks halfway (after approx. 6 weeks).|
Each part is introduced via one or more lectures, accompanied by some written material or reference to accessible material. Students work on assignments related to the topic and produce a written report on every part of the course. At the end of the course, students chose a topic to be presented by means of a poster during a seminar. Below we briefly summarize the topics of the various parts of the course.
Part 1 (2.2EC; approximately 4 weeks) deals with phase separations, starting from an analytical description, followed by a numerical treatment.
Part 2 (1.1EC; approximately 2 weeks) focuses on physics problems that can be mapped onto solving sets of (non-linear) equations numerically. One selection of problems revolves around solving the Poisson-Boltzmann equation. The numerical methods covered are the classical techniques of LU decomposition, fixed-point iteration, Jacobi and Gauss-Seidel iteration, and over-relaxation. A second set of problems focuses on the self-consistent polarization field and its effect on the transport gap in molecular crystals. Convergence acceleration is vital for solving this problem. For this, a special technique will be introduced, called Pulay iteration. The students will enjoy programming these algorithms and solving the physics problems themselves without much need for black-box routines.
Part 3 (1.1EC; approximately 2 weeks) focuses on physics problems that can be mapped onto (non-linear) eigenvalue equations. One example is the non-linear Schrödinger equation, which can be used to describe polaron particles in condensed matter, or soliton waves in non-linear optics. A second example is the constrained Schrödinger equation that describes the formation of a Cooper pair in a superconductor. A third example comprises the quantum rotations of the water molecule, which has implications for its thermodynamical properties, i.e. ortho-water and para-water. General numerical approaches for solving eigenvalue problems will be discussed and applied, such as (inverse) power iteration, Rayleigh quotient iteration and QR iteration. The students will enjoy programming these algorithms, learn to think inside the boxes, and solve some interesting physics problems.
Part 2 (1.1EC; approximately 2 weeks) focuses on the study of phase transitions. Techniques: Mean field, expansions in temperature, dimensions, degrees of freedom, renormalization group. General techniques: complex analysis, counting (loop expansions, generating functions, Pade)
Part 3 (1.1EC; approximately 2 weeks) focuses on the study of group theory and symmetry in physics. Techniques: determining groups and symmetries, Lagrangians, Euler-Lagrange, variational principles, gauge symmetry, Maxwell and weak/strong interactions, symmetry breaking, Noether’s theorem, Goldstone particles, Higgs particle.
Seminar (0.6EC): A poster will be prepared on one of the topics worked on during the course. During a closing seminar the poster will be presented to all students and teachers.
Three reports must be handed in. At the end there will be a poster presentation.
The final grade will be based on the three reports:
Report 1: 50%
Report 2: 25%
Report 3: 25%