MAJOR PHYSICS COURSE
Semester IV
MAJOR IV: PHYS4012: Mathematical Physics-II (Credits: Theory-04, Practicals-01)
F.M. = 75 (Theory-40, Practical–20, Internal Assessment–15)
Course Objectives: The emphasis of this course is on applications of mathematical techniques in solving problems of interest to physicists. Students are to be examined on the basis of problems, seen and unseen.
Fourier Series
Unit 1 Periodic functions, Orthogonality of sine and cosine functions, Dirichlet conditions (statement only), Expansion of periodic functions in a series of sine and cosine functions and the determination of Fourier coefficients, Complex representation of a Fourier series, Expansion of functions with arbitrary period, Expansion of non-periodic functions over an interval, Even and odd functions and their Fourier expansions. Applications: Summing of infinite series, Term-by-term differentiation and integration of a Fourier series, Parseval identity.
Frobenius Method and Special Functions
Singular points of the second order linear equations and their importance, Frobenius method and its applications to differential equations, Legendre, Bessel, Hermite and Laguerre differential equations.
Properties of Legendre polynomials: Rodrigues formula, Generating function, Orthogonality, Simple recurrence relations, Expansion of function in a series of Legendre Polynomials.
Bessel functions of the First kind: Generating function, Simple recurrence relations, Zeros of Bessel functions and orthogonality.
Dirac Delta Function:
Definition, Representation as a limit of a Gaussian function and a rectangular function, Properties of the Dirac delta function.
Some Special Integrals
Beta and Gamma functions and the relation between them, Expression of integrals in terms of Gamma functions, Error function (Probability integral).
Theory of Errors
Systematic and random errors, Propagation of errors, Normal law of errors, Standard and probable error.
Partial Differential Equations
Solutions to partial differential equations using the method of separation of variables: (1) the Laplace's equation in problems of rectangular, cylindrical and spherical symmetries, (2) the wave equation related to the vibration of a stretched string and the oscillations of membranes (rectangular and circular).
Introduction to Probability
Random experiments, Sample space, Events, Probability, Random variables and probability distributions: (1) Discrete distributions, Binomial distribution as an example (2) Continuous distributions Gaussian, and Poisson distribution as examples, Mean and Variance.
Complex Analysis
Brief revision of complex numbers and their graphical representation, Euler's formula, De Moivre's theorem, Roots of complex numbers, Functions of complex variables, Analyticity and Cauchy- Riemann conditions, Examples of analytic functions, Singular functions: poles and branch points, order of singularity, branch cuts, Integration of a function of a complex variable, Cauchy's Inequality, Cauchy’s Integral formula.
MAJOR IV: PHYS4012: Mathematical Physics-II
Practical: 30 Lectures
SCILAB [Course Syllabus]
Codes and Notes:
Introduction to Numerical Computation Software
Inverse of a Matrix, Eigen Vctors and Eigen Values
Solution of Ordinary Differential Equations
XCOS
Here is sample list of numerical experiments one can do.[here]