Methods Of Mathematical Physics
Course Content and Outline
Linear Vector Spaces (12 HOURS)
Linear operators - Representation of vectors and operators in a basis - Linear independence, dimension -Inner product - Schwarz inequality - Orthonormal basis -Gram-Schmidt Process.
Linear Differential Equations and Green’s Function (12 HOURS)
Second order linear differential equations - Sturm - Liouville Theory – Orthogonality of eigen functions - Illustration with Legendre, Laguerre, Hermite, and Chebyshev differential equations - Location of zeros of these polynomials - Wronskian, ordinary and singular points.
Green‘s function – Eigen function expansion of the Green‘s function - Reciprocity theorem - Sturm – Liouville type equations in one dimension and their Green‘s function.
Complex Variables (12 HOURS)
Functions of a complex variable - Single and multivalued functions - Analytic functions - Cauchy - Riemann conditions- Singular points - Cauchy‘s theorem and integral formulae- Taylor and Laurent expansions - Zeros and poles – Residue theorem and its applications.
Laplace and Fourier Transforms (12 HOURS)
Laplace transforms - Solution of linear differential equations with constant coefficients - Fourier integral - Fourier transforms, Fourier sine and cosine transforms - Convoltion theorems - Applications.
Group Theory (12 HOURS)
Basic definitions - Lagrange‘s Theorem – Invariant subgroup - Homomorphism and Isomorphism between groups - Representation of a group - unitary representations -Schur‘s lemmas - Orthogonality theorem - Character table- Simple applications to symmetry groups and molecular vibrations.
MODE OF DELIVERY
- Lectures and discussions
- Reading and problem assignments
INSTRUCTIONAL MATERIALS AND / OR EQUIPMENT
- Whiteboard and Markers
- Flip Charts
- LCD Projectors
- CDs, DVDs and Tapes