Frames of Reference: Inertial Frames, Galilean Transformations, Non-inertial Frames, Rotating Frames, Accelerated Frames; Equilibrium and Forces: Various Forces of Nature, Conservative Forces, Inertial and Non-inertial Forces, Frictional Forces, Central Forces; Inertia and Motion: Newton's laws of motion, Simple Harmonic Oscillator, Inverse Square Law; Conservation Laws: Energy, Linear and Angular Momentum, Collisions; Elementary Dynamics of Rigid Bodies; Newtonian Gravitation; Basic Special Relativity: Lorentz Transformations, Relativistic Dynamics, Momentum and Energy; The Principle of Equivalence; Generalised Coordinates, Principle of Least Action, An Elementary Introduction to Lagrangian and Hamiltonian Dynamics.
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Vector Calculus: Gradient, Divergence, and Curl, Gauss' Greens' and Stokes' Theorems; Electrostatics: Charges, Fields and Potentials, Magnetostatics: Currents, Fields and Potentials, Electromagnetic Induction, Displacement Current and Maxwell's Equations, Plane Electromagnetic Waves, Currents and Conductors, Electric and Magnetic Fields in Matter; Boundary Conditions at a Surface of Discountinuity; Conservation Law of Energy: Poynting's Theorem, Conservation Laws of Momentum and Angular Momentum, Maxwell's Stress Tensor.
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Introduction to basic programming principles using Python, including object-oriented design, big-oh notation, sorting and search algorithms, elementary data structures (lists, heaps, binary trees).
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Linear Algebra: General Linear Vector Spaces: Matrices, Special Matrices (symmetric, hermitian, orthogonal, unitary), Determinant, Rank, Inverse of a Matrix, Eigenvalue Problem, Orthogonalization Theorem, Matrix Diagonalization, Normal Matrices, Canonical Forms, Scalar Product, Dual Vectors, Cauchy-Schwarz Inequality, Real and Complex Vector Spaces, Metric Space; N-dimensional Vector Space: Change of Basis in an N-dimensional Space, Scalars, Cartesian Tensors, Tensor Notation and Operations, Inertia Tensor, Kronecker Delta, Levi-Civita Symbol, Pseudovectors and Pseudotensors. Ordinary Differential Equations: First-order ODEs, Separable ODEs, Exact ODEs, Integrating Factors, Linear ODEs, Existence and Uniqueness of Solutions, Second-order Linear ODEs, Homogeneous Linear ODEs with Constant Coefficients, Differential Operators, Free Oscillations, Euler-Cauchy Equations, Existence and Uniqueness of Solutions, Wronskian, Nonhomogeneous ODEs, Forced Oscillations, Resonance, Electric Circuits, Solution by Variation of Parameters, Higher-order ODEs, Systems of ODEs, Constant Coefficient Systems, Phase-plane Method, Criteria for Critical Points, Stability, Qualitative Methods for Nonlinear Systems, Nonhomogeneous Linear System of ODEs, Power Series Method, Legendre's Equation, Legendre Polynomials, Frobenius Method, Bessel's Equation and Bessel Functions of First and Second Kind, Sturm-Liouville Theory, Orthogonal Functions, Orthogonal Function Expansions.
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Scalar Waves: Plane waves, Spherical Waves, Harmonic Waves, Phase Velocity, Wavepackets, Group Velocity; Vector Waves: Elliptic, Linear and Circular Polarization, Stokes Parameters; Reflection and Refraction, Fresnel Formulae, Total Reflection; Elementary Dispersion Theory. Eikonal Approximation: Ray and Matrix Optics, Fermat's Principle, Optical Imaging; Aberrations: Chromatic, Spherical, Coma, Astigmatism, Distortion; Interference and Interferometers, Division of Wavefront, Division of Amplitude, Multiple-beam Interference; Elementary Theory of Diffraction: Kirchoff' theory, Fraunhofer and Fresnel Diffraction; Elementary Scattering Theory, Crystal Optics.
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Fourier Analysis: Fourier Series, Fourier Integral, Fourier Transform, Generalised Functions, Fourier Transform of a Generalised Function, the Dirac Delta Function, Green's Functions and Delta Functions, Green's Functions in Various Dimensions, Solving Differential Equations using Green's Functions, Integral Equations.
Partial Differential Equations: Laplace'e Equation, The Diffusion Equation, The Wave Equation, Poisson's Equation, Special Functions.
Complex Analysis: Complex Numbers, Complex Plane, Argand Representation, Powers and Roots, Derivative, Analytic Function, Cauchy-Riemann Equations, Laplace's Equation, Exponential Function, Trigonometric and Hyperbolic Functions, Logarithm, General Power; Line Integral in the Complex Plane, Cauchy's Integral Theorem, Cauchy's Integral Formula, Derivatives of Analytic Functions; Power Series, Taylor Series, Laurent Series, Calculus of Residues, Evaluation of Definite Integrals, Multivalued Functions, Conformal Mapping.
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The Physical Basis of Quantum Mechanics: Experimental Background, The Old Quantum Theory, Uncertainty and Complementarity, Discussion of Measurement, Wave Packets in Space and Time; The Schrodinger Equation: Development of the Wave Equation, Interpretation of the Wave Function, Energy Eigenfunctions, One-dimensional Square Well Potential; Eigenfunctions and Eigenvalues: Interpretative Postulates and Energy Eigenfunctions, Momentum Eigenfunctions, Motion of a Free Wave Packet in One Dimension; Discrete Eigenvalues (Bound States): Linear Harmonic Oscillator, Spherically Symmetric Potentials in Three Dimensions, Three-dimensional Square Well Potential, The Hydrogen Atom; Continuous Eigenvalues (Collision Theory): One-dimensional Square Well Potential, Collisions in Three Dimensions, Scattering by Spherically Symmetric Potentials, Scattering by Complex Potentials, Scattering by a Coulomb Field.
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The laws of Thermodynamics: First law, Second law, Entropy, Thermodynamic Potentials, Third law; Applications of Thermodynamics: Description of Phase Transitions, Surface Effects in Condensation, Van der Waals Equation of State, Osmotic Pressure; Probability: General Definitions, One Random Variable, Some Important Probability Distributions, Many Random Variables; Kinetic Theory: Binary Collisions, Boltzmann Transport Equation, Boltzmann's H Theorem, Maxwell-Boltzmann Distribution, Most Probable Distribution; Transport Phenomena: Mean Free Path, Conservation Laws, The Zeroth-order Approximation, the First-order Approximation, Viscosity, Viscous Hydrodynamics, The Navier-Stokes Equation, Examples in Hydrodynamics.
Advanced Topics: Sums of Random Variables and the Central Limit Theorem, Rules for Large Numbers, Information, Entropy, and Estimation.
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Lagrangian Dynamics: Generalised Coordinates, Principle of Least Action, Lagrange's Equations of Motion for One Particle and for Systems of Particles; Conservation Laws: Energy, Momentum, Centre of Mass, Angular Momentum; Motion in One Dimension, Motion in a Central Field, Kepler's Problem; Collisions: Disintegration of Particles, Elastic Collisions, Scattering, Rutherford's Formula, Small Angle Scattering; Small Oscillations: Free Oscillations, Forced Oscillations, Vibrations of Molecules, Damped Oscillations, Forced Oscillations Under Friction, Parametric Resonance, Anharmonic Oscillations, Resonance in Non-linear Oscillations, Motion in a Rapidly Oscillating Field; Rigid Body Dynamics: Angular Velocity, The Inertia Tensor, Angular Momentum of a Rigid Body, The Equations of Motion of a Rigid Body, Eulerian Angles, Euler's Equations, The Asymmetrical Top, Rigid Bodies in Contact, Motion in a Non-inertial Frame of Reference; The Canonical Equations: Hamilton's Equations, The Routhian, Poisson Brackets, Maupertius Principle, Canonical Transformations, Liouville's Theorem, The Hamilton-Jacobi Equation, Adiabatic Invariants, Canonical Variables.
Advanced Topics: Nonlinear Oscillations: Stability of Solutions, The Poincare-Bendixon Theorem, The Poincare Map, Logistic Map, The Circle Map, Chaos in Hamiltonian Systems and the KAM Theorem.
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Special Theory of Relativity: Experimental Basis, Lorentz Transformations and Basic Kinematic Results, Addition of Velocities, 4-Velocity, Relativistic Momentum and Energy of a Particle, Applications of Relativity to Optics, Thermodynamics, Elasticity and Fluids, Spacetime of Special Relativity, Matrix Representations of Lorentz Transformations, Infinitesimal Generators, Thomas Precession, Covariance of Electrodynamics, Transformation of Electromagnetic Fields; Lagrangian and Hamiltonian for a Relativistic Charged Particle in External Electromagnetic Fields, Motion in a Uniform Static Magnetic Field, Motion in Uniform, Static, Electric and Magnetic Fields, Particle Drifts in Nonuniform, Static Magnetic Fields; The Darwin Lagrangian, Lagrangian of the Electromagnetic Field, The Proca Lagrangian, Photon Mass Effects; Canonical and Symmetric Stress Tensors, Conservation Laws, Solution of the Wave Equation in the Covariant Form; Radiation by Moving Charges, Lienard-Wiechert Potentials, Larmor Formula and Its Relativistic Generalization, Angular Distribution of Emitted Radiation, Frequency Spectrum of Emitted Radiation, Bremsstrahlung; Radiation Damping.
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Basic Group Theory: Definitions and Examples, Subgroups, The Symmetric Group, Classes and Invariant Subgroups, Cosets and Factor (Quotient) Groups, Homomorphisms, Direct Products; Group Representations, Irreducible, Inequivalent Representations, Unitary Representations, Schur's Lemmas, Orthonormality and Completeness Relations of Irreducible Representation Matrices, Characters, Regular Representation, Direct Product Representation, Clebsch-Gordon Coefficients; Irreducible Basis Vectors, Projection Operators, Wigner-Eckart Theorem; Representations of the Symmetric Groups, Young Diagrams; Continuous Groups: SO(2), SO(3), and SU(2), Euclidean Groups in Two and Three Dimensions, The Lorentz and Poincare Groups, Space-Time Symmetries, Parity and Time-Reversal Invariance. Numerical Analysis: Solution by Iteration, Finite Difference, Interpolation, Numerical Integration and Differentiation, Asymptotic Expansions; Numerical Methods in Linear Algebra: Gauss Elimination, Matrix Inversion, Iteration, Least Squares; Numerical Methods in Differential Equations, Optimization.
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Matrix Formulation of Quantum Mechanics: Matrix Algebra, Transformation Theory, Equations of Motion; Symmetry in Quantum Mechanics, Space and Time Displacements, Rotation, Angular Momentum and Unitary Groups, Combination of Angular Momentum States and Tensor Operators, Space Inversion and Time Reversal, Dynamical Symmetry; Approximation Methods for Bound States: Stationary Perturbation Theory, Variational Method, Dalgarno-Lewis Method, WKB Approximation, Time-dependent Perturbation Theory; Approximation Methods in Collision Theory: The Scattering Matrix, Stationary Collision Theory, Born Approximation, Distorted Wave Born Approximation, Partial Wave Analysis, Eikonal Approximation, Analytic Properties and Dispersion Relations; Identical Particles and Spin, Density Operator and Density Matrix, Rearrangement Collisions, T matrix, Creation and Annihilation Operators, The Algebra of Creation and Annihilation Operators, Dynamical Variables, Continuous One-Particle Spectrum and Quantum Field Operators, Quantum Dynamics of Identical Particles and Second Quantization.
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The Postulate of Classical Statistical Mechanics, General Definitions, The Microcanonical Ensemble, Two-level Systems, The Ideal Gas, Mixing Entropy and the Gibbs Paradox, The Canonical Ensemble, Canonical Examples. The Gibbs Canonical Ensemble, The Grand Canonical Ensemble, The Equivalence of the Canonical and the Grand Canonical Ensemble; Interacting Particles: The Cumulant Expansion, The Cluster Expansion, The Second Virial Coefficient and the van der Waal's Equation, Breakdown of the van der Waal's Equation, Mean Field Theory of Condensation, Variational Methods, Corresponding States, Critical Point Behaviour; Quantum Statistical Mechanics: The Postulates of Quantum Statistical Mechanics, Density Matrix, Ensembles in Quantum Statistical Mechanics, Dilute Polyatomic Gases, Vibrations of a Solid, Black-body Radiation; Ideal Gases: The Ideal Fermi Gas, Equation of State, Theory of White Dwarfs, Landau Diamagnetism, De Haas-Van Alphen Effect, Pauli Paramagnetism, The Ideal Bose, Photons, Phonons, Bose-Einstein Condensation, Imperfect Fermi Gas, Imperfect Bose Gas; Statistical Mechanical Theory of Phase Transitions: Ising Model, Lattice Gas, Broken Symmetry and Range of Correlations, Mean Field Theory, Renormalization Group Theory, Isomorphism Between Two-Level Quantum Mechanical System and the Ising Model.
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Advanced Topics: The Hartree-Fock Method, and the Self-Consistent Field Method; Theory of Multiplets: Electrostatic Interaction, Spin-Orbit Interaction, Interactions with External Fields; Atomic Collisions: Elastic Scattering At High Energies, At Low Energies, Corrections to Elastic Scattering, Elastic Scattering of Spin 1/2 Particles, Inelastic Scattering At High Energies, At Low Energies, Semi-classical Treatment of Inelastic Scattering, Classical Limit of Quantum Mechanical Scattering.
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Advanced Topics: Radiative Corrections, Photon Self-Energy, Electron Self-Energy, Vertex Modification, Lamb Shift, Regularization, Vacuum Polarization, Anomalous Magnetic Moment, Renormalization of QED.
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Particles and Interactions, Gauge Theories: Internal Symmetries, Isospin, Unitary Symmetry, Representation of SU(3); The Quark Model, Color, Evidence for Color, Parton Model, Bjorken Scaling; Charm, the Charmed Quark, J/Psi and its Family, Correspondence between Quarks and Leptons; PCAC and Soft Pion Theorems; The Vector-Current Ward Identity, Axial-Vector-Current Ward Identity, Anomaly; Phenomenology of Weak Interactions: The Weinberg-Salam Model, GIM Mechanism, W and Z Particles, The Higgs Particle, CP Violation, CKM Mixing.
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