## Details of PH3103 (Autumn 2019)

 Level: 3 Type: Theory Credits: 4.0

Course CodeCourse NameInstructor(s)
PH3103 Mathematical Methods of Physics Ritesh Kumar Singh

Preamble
This is a first course on the mathematical methods of physics. This will introduce students to the mathematical techniques which will be essential to a study of physics. The emphasis will be more on techniques rather than on rigour. Can be taken by all 3rd year BS-MS Students.

Syllabus

• Linear algebra: Recapitulation of basic concepts. Dual spaces, Eigenvalues, Eigenvectors, Similarity transformations, Diagonalization, Inner product spaces, Hilbert spaces.

• Complex analysis: Differentiability and analyticity, the Cauchy-Riemann conditions, the Cauchy theorem, Cauchy's residue theorem, Applications to the calculation of integrals and sums.

• Fourier and Laplace transforms: inversion of Fourier and Laplace transforms, applications to the solution of differential equations.

• Sturm-Liouville systems: Orthogonal polynomials, The hypergeometric and confluent hypergeometric functions, Bessel, Neumann and Hankel functions, Legendre and Hermite polynomials, Integral representations of special functions.

• Green's functions: For first and second order linear differential equations in one dimension, eigen-function expansions, connection with the delta function, integral equations, perturbation theory based on Green's function.

• Group theory for the physical sciences: Review of the algebra of groups. Action of a group on a set. Action on function spaces. Matrix groups. Matrix representations. Reducible and irreducible representations. Schur's Lemma. Characters. Orthogonality relations. Regular representation. Character tables. Reduction of representations. Applications to crystallography, molecular vibrations and molecular orbital theory.

• Continuous and Lie groups: Irreducible representations of the groups SO(3) and SU(2). Infinitesimal group elements. Lie algebras. Simple and semi-simple Lie groups and algebras. The Cartan subalgebra. Roots and weights. Classification of simple root spaces. Dynkin diagrams. Irreducible representations of su(3) - the quark model.

Prerequisite
Mathematical method courses in Levels 1 and 2; Mentor guidance for 1st year IPhD students.

References

1. G. B. Arfken and H. J. Weber, Mathematical methods for physics, Academic press (2005).

2. E. Kreyszig, Advanced engineering mathematics, Wiley eastern (1998).

3. S. Hassani, Mathematical physics : a modern introduction to its foundations, Springer-Verlag (1999).

4. P. Bamberg and S. Sternberg, A course in mathematics for students of physics Vol 1 , Cambridge university press (1998).

5. Group theory in physics and chemistry, Springer-Verlag (2005).

6. S. Sternberg, Group theory and physics, Cambridge University Press (1994).

7. R. Gilmore, Lie groups, Lie algebras, and some of their applications, John Wiley and Sons, New York (1974).

#### Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 1 Core
2 IP 3 Not Allowed
3 IP 5 Not Allowed
4 MR 1 Elective
5 MR 3 Not Allowed
6 MS 5 Core
7 RS 1 Elective
8 RS 2 Elective