Methods Of Mathematical Physics
A comprehensive problem solving oriented course designed for students in physics programs. Various mathematical methods as applied to the relevant problems in the physical sciences are discussed. Topics: Linear Vector Spaces, Linear Differential Equations and Green‘s Function, Complex Variables, Group Theory and Laplace and Fourier Transforms. Three hours lecture and one hour computer laboratory, utilizing the symbolic/numerical/graphical package.
To provide concepts and practice in, mathematical methods commonly applied to the study and practice of physics.
By the end of this course, the student should be able to:
- Explain the theory of residues and contour integration to evaluate non-simple integrals.
- Apply the Fourier transforms to solve linear partial differential equations of the infinite and semi-infinite real axis. Use the residue theorem to calculate inverse Fourier transform.
- Define linear vector space in general and the Banach and Hilbert space in particular.
- Discuss suffix notation for vectors and tensors; apply the transformation rules for vectors, tensors and pseudo-tensors.
- Explain the fundamental techniques of classical and (especially) quantum mechanics at a theoretically sophisticated level,
- Demonstrate usefulness of this techniques in 'real world' problems, thereby helping students to master other core subjects (classical mechanics, electrodynamics, quantum mechanics among others)
A student completing the course is expected to:
- Describe physical systems using variation of parameters.
- Use and manipulate functions defined as integrals.
- Manipulate and evaluate functions of complex variables.
- Apply vector and tensor calculus to physical systems.
- Describe physical systems in terms of curvilinear coordinates.
- Describe and evaluate physical systems in terms of eigenvalues, eigenvectors, tensors, and matrices.
- Apply group and operator theory to systems in classical and quantum mechanics.
- Apply Hamiltonian and Lagrangian formalisms of classical mechanics to physical systems.
- Use classical scattering theory to describe particle interactions.
- Apply Green’s function methods to problems in classical and quantum mechanics, and in electromagnetism.
- Describe physical systems in terms of partial differential equations. Determine solutions to first and second order partial differential equations.
- Apply complex variable theory to physical systems, including Laurent series, conformal mapping, Laplace transforms, and the evaluation of real integrals with the calculus of residues.