
The learning objectives are summarized below:
 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.
Emphasis in the learning objectives is on the physical interpretation of the results obtained with the mathematical and/or numerical models


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.
Numerical track:
Part 2 (1.1EC; approximately 2 weeks) focuses on physics problems that can be mapped onto solving sets of (nonlinear) equations numerically. One selection of problems revolves around solving the PoissonBoltzmann equation. The numerical methods covered are the classical techniques of LU decomposition, fixedpoint iteration, Jacobi and GaussSeidel iteration, and overrelaxation. A second set of problems focuses on the selfconsistent 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 blackbox routines.
Part 3 (1.1EC; approximately 2 weeks) focuses on physics problems that can be mapped onto (nonlinear) eigenvalue equations. One example is the nonlinear Schrödinger equation, which can be used to describe polaron particles in condensed matter, or soliton waves in nonlinear 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. orthowater and parawater. 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.
Analytical track:
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, EulerLagrange, 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.
Assessment plan
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%
Seminar/poster: Pass/Fail





  Required materialsRecommended materialsHandouts 
 Instructional modesAssignmentPresence duty   Yes 
 Lecture
 TutorialPresence duty   Yes 

 TestsReports and poster session


 