# Scheme m. Sc. Physics Nano–Science & Technology

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## SECTION A

Gamma and Beta functions: Definition and their relations
Bessel functions: Series solutions of Bessel's differential equation recurrence relations, Evaluation of Jn(x) for half-integral, generating function, Orthogonality (statement only).
Legendre Polynomials: Series solution of Legendre differential equation, Rodrigue and recurrence formulae, Generating function; Associated Legendre equation and polynomials;
Hermite polynomials: Series solution of Hermite differential equation, Hermite polynomials, Generating functions, Recurrence relations, Orthogonality (statement only), Simple integral involving Hermite polynomials.
Laplace transforms, Definition, Laplace transform of elementary functions, Basic theorems of Laplace transforms, Inverse Laplace transforms, its properties and related theorems, Convolution theorem, Use of Laplace transforms in the solution of differential and integral equations, Evaluation of integrals using Laplace transforms.
Fourier series and transform: Dirichlet conditions, Expansion of periodic functions in Fourier series, Sine and cosine series, The finite Fourier sine and cosine transforms, Complex form of Fourier series, Fourier integral theorem and Fourier transform, Parseval's identity for Fourier series and transforms.

SECTION B
Partial differential equations, One dimensional wave equation, The vibrating string fixed at both ends, D'Alembert and Fourier series solutions, Vibrations of a freely hanging chain, Two dimensional wave equation in rectangular membrane, Wave equation in the two dimensional polar coordinates and vibrations of a circular membrane, 3-D wave equation and its solution, Equation of heat conduction, Two dimensional heat conduction, Temperature distribution in a rectangular and circular plate, 3-D heat conduction equation.

Evaluation of polynomials: Horner's method; Root finding: Fixed point iteration, Bisection method, Regula falsi method, Newton method, Error analysis; System of linear equations: Gauss elimination, Gauss Seidel method, Interpolation and Extrapolation: Lagrange's interpolation, least square fitting;

Differentiation and Integration: Difference operators, Simpson and trapezoidal rules; Ordinary differential equation: Euler method, Taylor method.

Text Books:

1. Applied Mathematics: L.A. Pipes and Harwill, Mc Graw Hill Publication

2. Mathematical Physics: G.R.Arfken, H.I.Weber, Academic Press, USA (Ind.Ed.)

3. Laplace Transforms: M.R. Speigel (Schaum Series), Mc Graw Hill Publication

4. Numerical Methods: J.H. Mathews, Prentice Hall of India, New Delhi

Reference Books:

1. Advanced Engg. Mathematics: E. Kreyszig, Wiley Eastern Publication

## NT 1.1.2CLASSICAL MECHANICS

Maximum Marks: External 60 Time Allowed: 3 Hours

Internal 20 Total Teaching hours: 50

Total 80 Pass Marks: 35%

Out of 80 Marks, internal assessment (based on two mid-semester tests/ internal examinations, written assignment/project work etc. and attendance) carries 20 marks, and the final examination at the end of the semester carries 60 marks.
Instruction for the Paper Setter: The question paper will consist of three sections A, B and C. Each of sections A and B will have four questions from respective section of the syllabus. Section C will have 10 short answer type questions, which will cover the entire syllabus uniformly. Each question of sections A and B carries 10 marks. Section C will carry 20 marks.
Instruction for the candidates: The candidates are required to attempt two questions each from sections A and B, and the entire section C. Each question of sections A and B carries 10 marks and section C carries 20 marks.
Use of scientific calculators is allowed.

## Section A

Lagrangian Formulation: Conservation laws of linear momentum, angular momentum and energy for a single particle and system of particles, Constraints and generalized co-ordinates, Principle of virtual work, D' Alembert's principle and Lagrange's equations of motion, for conservative systems. Applications of Lagrangian formulation.

Variational Principle: Hamilton's principle, Calculus of variations and its application to the shortest distance, minimum surface area of revolution and the brachistochrone problem. Lagrange's equations from Hamilton's principle. Generalized momentum, Cyclic co-ordinates, Symmetry properties and Conservation theorems.

Two body Central Force Problem: Equivalent one body problem, Equations of motion and first integrals, Classification of orbits, Differential equation for the orbit, Kepler problem, Differential and total scattering cross-section, Scattering in an inverse square force field and Rutherford scattering cross section formula, Scattering in lab and center of mass frame.

Section B

Hamiltonian Formulation: Legendre transformation, Hamilton's equations of motion and their physical applications, Hamilton's equations from variational principle, Principle of least action.

Canonical Transformations: Point and canonical transformations, Generating functions, Poisson's brackets and its canonical invariance, Equations of motion in Poisson Bracket formulation, Poisson bracket relations between components of linear and angular momenta. Harmonic oscillator problem, check for transformation to be canonical and determination of generating functions.

Small Oscillations: Eigen value equation, Frequencies of free vibration and normal modes, Normal mode frequencies and eigen vectors of diatomic and linear tri-atomic molecule.

Rigid Body Motion: Orientation of a rigid body, Orthogonal transformations and properties of the orthogonal transformation matrix, Euler angles, Euler's theorem, Infinitesimal rotation, Rate of change of vector in rotating frame, Components of angular velocity along space and body set of axes. Motion of heavy symmetrical top (Analytical treatment).

Text Books:

1. Classical Mechanics: H. Goldstein (Narosa Pub.)

2. Classical Mechanics: J.C. Upadhyaya (Himalaya Pub. House)

NT 1.1.3 CONDENSED MATTER PHYSICS
Maximum Marks: External 60 Time Allowed: 3 Hours

Internal 20 Total Teaching hours: 50

Total 80 Pass Marks: 35%
Out of 80 Marks, internal assessment (based on two mid-semester tests/ internal examinations, written assignment/project work etc. and attendance) carries 20 marks, and the final examination at the end of the semester carries 60 marks.
Instruction for the Paper Setter: The question paper will consist of three sections A, B and C. Each of sections A and B will have four questions from respective section of the syllabus. Section C will have 10 short answer type questions, which will cover the entire syllabus uniformly. Each question of sections A and B carries 10 marks. Section C will carry 20 marks.
Instruction for the candidates: The candidates are required to attempt two questions each from sections A and B, and the entire section C. Each question of sections A and B carries 10 marks and section C carries 20 marks.
Use of scientific calculators is allowed.

SECTION A
Diffraction methods, Lattice vibrations, Free electrons: Diffraction methods, Scattered wave amplitude, Reciprocal lattice, Brillouin zones, Structure factor, Quasi Crystals, Form factor and Debye Waller factor, Bonding of solids, Lattice vibrations of mono-atomic and diatomic linear lattices, IR absorption, Neutron scattering, Free electron gas in 1-D and 3-D. Heat capacity of metals, Thermal effective mass, Drude model of electrical conductivity, Wiedman-Franz law, Hall effect, Quantized Hall effect.