Syllabus

Sketching functions, Gaussian integrals, Stirling's formula, Generalised functions: Step function, Dirac delta function, properties of delta function,

Vectors and Tensors: Cartesian tensors, covariant and contravariant components of a vector, covariant and contravariant tensor, mixed tensor, metric tensor, contraction, rotations and index notation, Isotropic tensors: Kronecker delta, Levi-Civita symbol, Gram determinant. Rotations in three dimensions, Proper and improper rotations, scalars and pseudoscalars; polar and axial vectors.

Linear vector spaces: Definitions and basic properties, the dual of a linear space, the inner production of two vectors, basis sets and dimensionality, Gram-Schmidt orthonormalisation, Expansion of an arbitrary vector, Basis-independence of the inner product, The Cauchy-Schwarz inequality

Matrices: Pauli matrices, Expansion of a (2x2) matrix, the exponential of a matrix, Rotation matrices in three dimensions: generators of infinitesimal rotations and their algebra, matrices as operators in a linear space, projection operators, Hermitian, unitary and positive definite matrices.

Infinite-dimensional vectors spaces: square-summable sequences, square-integrable functions, continuous basis, wave function of a particle, Hilber space, subspaces, linear manifolds.

Linear operators on a vector space: linear operators, norm and bounded operators, adjoint of an operator, derivative operator in square integrable space, adjoint of the derivative operator, nonsymmetric operators, matrix representations of unbounded operators. Useful operator identities: Hadamard's Lemma, Zassenhaus formula, Baker- Campbell-Hausdorff formula.

Orthogonal polynomials: Orthogonality and completeness, recursion relation, the classical orthogonal polynomials, hypergeometric differential equation, Rodrigues formula and generating function, Hermite polynomials, linear harmonic oscillator eigenfunctions, generalized Laguerre polynomials, Jacobi polynomials:Chebyshev polynomials and Legendre polynomials, associated Legendre functions, Spherical

harmonics,

Discrete probability distributions, mean and variance, Bernoulii trials and the binomial distribution, number fluctuation in a classical ideal gas, the geometric distribution, photon number distribution in blackbody radiation, The Poisson distribution, Photon number distribution in coherent radiation, The sum of Poisson-distributed random variables, The simple random walk

Continuous probability distributions: Probability density and cumulative distribution, The moment-generating function, the cummulant-generating function, The Gaussian distribution, The Gaussian as a limit law, The central limit theorem.

Group Theory: Definitions and examples, discrete groups, cyclic groups, Subgroups, Cosets, Lie groups, Lie algebra and applications. Angular momentum operators, representation of rotations by SU(2) matrices, Connection between the groups SO(3) and SU(2), The parameter spaces of SU(2), SO(2) and SO(3). Isomorphism between SO(2) and U(1), The squeezing operator and the group SU(1,1), SU (1,1) generators in terms of Pauli matrices.