Department of Mathematics

: Mathematics; Subjects; 24 October 2024; Hits: 157

AMAT 41413: Quantum Mechanics and Quantum Field Theory

Course Code : AMAT 41413

Title : Quantum Mechanics and Quantum Field Theory

Pre-requisites : AMAT 11223, PMAT 21263

Learning Outcomes:

At the end of this course, the student should be able to

1. understand and explain the differences between classical and quantum mechanics

2. solve time-dependent and time-independent Schrodinger equation for some basic, physically important types of potentials

3. interpret the wave function and apply operators to it obtain information about particle’s physical properties such as position, momentum and energy

4. understand the role of uncertainty in quantum physics and use the commutation relations of operators to determine whether or not two physical properties can be simultaneously measured

5. relate the matrix formalism to the use of basis states, and solve simple problems in that formalism

6. demonstrate an understanding of field quantization and the expansion of the scattering matrix

7. understand relativistic wave equations 8. describe spin zero and spin half fields.

Course Contents:

Quantum Mechanics: Introduction to Quantum mechanics: Review of Finite systems and commutator brackets, Quantum mechanics in Hilbert space, Axiomatic structure of quantum mechanics, Heisenberg and interaction picture, Complete set of observables, Observables and their expectation values.

Schrodinger’s Equation: Schrodinger’s time -independent and time -dependent equations in one dimension, potential well, potential barrier and tunnelling, simple harmonic oscillator, Schrödinger’s equation in three dimensions, particle in a three dimensional box, Hydrogen atom, Completely continuous operators, uncertainty principle, scattering theory of two particles, potential scattering

Quantum Field Theory: Quantization, Relativistic wave equation, Spin zero fields and spin ½ fields, Momentum and angular momentum operators, Phase factor, Conventions between the spinners, Two - component theory.

Method of Teaching and Learning: A combination of lectures and tutorial discussions.

Assessment: Based on tutorials, tests and end of course examination.

Recommended Reading :

1. Schiff, L.I. (4th Ed., 2014). Quantum Mechanics, McGraw-Hill India.

2. Prugovecki, E. (2nd Ed., 2013). Quantum Mechanics in Hilbert Space, Dover Publication.

3. Liboff, R.L. (4th Ed., 2011). Introductory Quantum Mechanics, Pearson India.

4. Baggot, J. (1997). The meaning of Quantum Theory, Oxford University Press.

5. Sakurai, J.J. (2013). Advanced Quantum Mechanics, Replica Press (P) LTD, India.

6. Lee, T.D. (1981). Particle Physics and Introduction to Field Theory, Harwood Academics.

: Mathematics; Subjects; 24 October 2024; Hits: 30

AMAT 41433: Boundary Value Problems

Course Code : AMAT 41433

Title : Boundary Value Problems

Pre-requisites : PMAT 42373, PMAT 32322

Learning Outcomes:

After the completion of this course unit, the student will be able to:

1. classify partial differential equations

2. transform boundary value problems to other coordinate systems

3. solve two dimensional and three-dimensional boundary value problems in real life

4. apply Green’s theorem in boundary value problems

5. identify proper boundary conditions

6. apply method of images in boundary value problems.

Course Contents:

Partial Differential Equations in two variables: Linear second order equations in two independent variables, Normal forms, Hyperbolic, parabolic and elliptic equations, Boundary value problems in rectangular and cylindrical coordinates, Applications to heat flow, Vibrations and waves, Laplace and Poisson equations in two dimensions.

Boundary Value Problems in Three Dimensions: Green’s theorem in three dimensions, Uniqueness of solutions with Dirichlet and Neumann boundary conditions , Formal solutions of Boundary value problems in electrostatics, Method of images, Laplace equation in spherical polar coordinates, Boundary value problems with spherical symmetry, heat and wave equations , Three dimensional boundary value problems with azimuthal symmetry , Legendre functions and applications to gravitation and electrostatics, Potentials of circular rings and discs.

Method of Teaching and Learning: A combination of lectures and tutorial discussions.

Assessment: Based on tutorials, tests and end of course examination.

Recommended Reading :

Hebernsann, R. (1987). Elementary Applied Partial Differential Equations, Prentice-Hall.
Raisinghania, M.D. (1991). Ordinary and Partial Differential Equations, S. Chands, India.
Zill, D.G. & Cullen, M.R. (2018). Differential Equations with Boundary Value Problems, Cengage.

: Mathematics; Subjects; 27 February 2020; Hits: 1732

AMAT 42853: Tensor Analysis and General Relativity

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Course Code : AMAT 42853

Title : Tensor Analysis and General Relativity

Pre-requisites : PMAT 21553

Learning Outcomes:

At the end of the course the student should be able to demonstrate knowledge of concepts and theorems in tensor algebra, tensor analysis and the formalism of general relativity, and to solve Einstein’s field equations in simple cases, and to apply the solutions in Astrophysics.

Course Contents:

Covariant and contravariant vectors, Metric tensor, Invariants, Inner products, Differential-forms, Tensor analysis, Covariant differentiation of tensors, Einstein’s field equations, Schwarzchild interior and exterior solutions, Rotating systems, Dragging of inertial frames, Gravitation red shift, Bending of light and gravitational lenses.

Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

Assessment: Based on tutorials, tests, presentations and end of course examination.

Recommended Reading:

Adler R., Bazin M. and Schiffer M., Introduction to General Relativity, McGraw Hill, New York. (1975).
Misner, C.W.,Thorne, K.S. & Wheeler, J.A., Gravitation, Princeton University Press, 2017.
Hawking S.W. & Ellis G.F.R., The Large Scale Structure of Space-time, Cambridge University Press. (1975).
James B. Hartle, Gravity: An Introduction to Einstein's General Relativity, Pearson (2013).

: Mathematics; Subjects; 27 February 2020; Hits: 208

PMAT 32342: Number Theory

Course Code : PMAT 32342

Title : Number Theory

Pre-requisites : PMAT 12242

Learning Outcomes:

After the completion of this course unit, the student will be able to:
1. demonstrate a foundational understanding of number theory, including the definitions, conjectures, and
theorems that permit exploration of topics in the field.
2. work with numbers and polynomials modulo a prime, linear congruences, and systems of linear
congruences, including their solution via the Chinese remainder theorem.
3. Apply mathematical ideas and concepts within the context of number theory.
4. Solve a range of problems in number theory.
5. demonstrate an understanding of the mathematical underpinning of cryptography.

Course Contents:

Divisibility: Division and Linear Diophantine Equations, Greatest common divisor, Euclidean Algorithm, Prime
and Composite Numbers, Fibonacci and Lucas Numbers
Congruences: Linear Congruences, Chinese remainder theorem, systems of linear congruences,
Applications of Congruences: Divisibility tests, Round-robin tournament, Special Congruences: Wilson’s
theorem, Fermat’s little theorem, Euler’s theorem
Multiplicative Functions: Euler’s phi-functions
Primitive Roots: order of an integer and primitive roots, Primitive roots for primes, Primality testing using
primitive roots, Perfect Numbers
Cryptology: Affine Ciphers, Hill Ciphers, Exponentiation Ciphers.
Quadratic Residues and Reciprocity: Quadratic Residues, The Legendre Symbols, Quadratic Reciprocity

Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

Assessment: Based on tutorials, tests, presentations and end of course examination.

Recommended Reading:

1. K. Rosen, Elementary Number Theory and its Applications (6th Edition), Pearson (2010).
2. T. Koshy, Elementary Number Theory with Applications (2 nd Edition),‎ Academic Press (2007)
3. G. Andrews, Number Theory, Dover Publications (1994)
4. O. Ore, Number Theory and Its History, Dover Publications (1988)
5. W. Trappe, L. Washington, Introduction to Cryptography with Coding Theory (2 nd Edition), Pearson
International Press (2006)

: Mathematics; Subjects; 27 February 2020; Hits: 229

PMAT 42443: Advanced Topics in Geometry

Course Code : AMAT 42443

Title : Advanced topics in Geometry

Pre-requisites : PMAT 22293

Learning Outcomes:

After the completion of this course unit, the student will be able to:
1. develop an intuitive understanding of the nature of general equation of second degree
2. apply transformations and use symmetry to analyze mathematical situations
3. define conic sections and their properties
4. explain and use various techniques for calculating tangents, normal, pair of tangents, pole and polar of
conic sections
5. compute lines, plane, cone, sphere, ellipsoid and hyperboloid in three dimensions
6. analyze characteristics and properties of two dimensional and three-dimensional geometric shapes and
develop mathematical arguments about geometric relationships
7. demonstrate the knowledge of geometry and its applications in the real world.
8. develop knowledge and intuitive understanding on projective geometry.

Course Contents:

Analytical Geometry in Two Dimensions: Pairs of straight lines, General equation of second degree,
Translations and rotations of axes, Invariants of transformations
Conic sections: Classifying general equation of second degree into Parabolas, Ellipses, Hyperbolas, eccentricity.
Equation of tangent, Pairs of tangents and chords of contact, Harmonic conjugates, Pole and Polar, Parametric
treatment, Degenerate conic, Properties of conic.
Analytical Geometry in Three Dimension: Coordinate system, Direction cosines and direction ratios of a line,
Angle between two lines, Parallel lines, Perpendicular lines, Plane and Straight line, Shortest distance between
two-non intersecting lines, Skewed lines, General Equation of the second degree, Sphere, Cone, Ellipsoid and
hyperboloid, Tangent plane, Normal, Pole and Polar.
Projective Geometry: Projective Spaces; Definitions and Properties, Hyperplane at Infinity, Projective lines;
Projective Transformation of Projective Line Space, The Cross Ratio, Introduction to projective plane

Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

Assessment: Based on tutorials, tests, presentations and end of course examination.

Recommended Reading:

1. Chatterjee, D., (2008) Analytical Geometry, Alpha Science International.
2. Thomas, G.B. and Finney, R.L., (2008) Calculus and Analytic Geometry.
3. Maxwell, E.A., (1962) Elementary Coordinate Geometry, Oxford University press.
4. Jain, P.K. & Ahmad, K. (1994) Analytical Geometry of Two Dimensions, Wiley.
5. Kishan, H. (2006) Coordinate Geometry of Two Dimensions. Atlantic Publishers & Dist.
6. Jain, P.K. (2005) A Textbook of Analytical Geometry Of Three Dimensions. New Age International.
7. McCrea, W.H. (2006). Analytical Geometry of Three Dimensions, Dover Publications, INC.
8. Richter-Gebert J. (2011) Perspectives on Projective Geometry: A Guided Tour Through Real and
Complex Geometry, Springer.

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