Applied mathematics for physicists

• Determine whether an operator and an equation is linear or nonlinear and understand the difference between the two types.
• Solve linear partial differential equations with Fourier transformation and understand the relation between a function and its Fourier spectrum.
• Solve linear ordinary differential equations with Green’s function.
• Derive the properties of Sturm-Liouville operators and Sturm-Liouville eigenvalue problems.
• Solve linear partial differential equations using separation of variables.
• Solve singular linear ordinary differential equations using Frobenius’ method.
• Know Bessel’s equations and be able to use Bessel functions.
• Derive the Euler-Lagrange equations and understand the concept of calculus of variations and Hamilton’s Principle for multi-dimensional systems with several variables.
• Apply the Rayleigh-Ritz variational technique to find ground state eigenfunctions and eigenvalues.
• Apply the collective coordinate variational approach to find and analyse solutions to nonlinear partial differential equations.

Fourier transformation and general integral transformations. Sturm-Liouville problems and special functions. Bessel functions. Green’s function. Calculus of variations.

In the 4 larger projects you will use the techniques you have learned to analyse Bose-Einstein Condensates, optical fibers, electromagnetic wave propagation, nonlinear crystals, laser beams, and free electrons.

01005/10036/10020

13 morning sessions (8-12) beginning with 3 lectures (of 35 minutes each) and group work (2 hours) about 4 projects.

Textbook:
Arfken & Weber: Mathematical Methods for Physicists, 6th Ed., Elsevier.