Professor Academy is offering a 50% concession on courses for PWD and Transgender aspirants
Professor Academy is offering a 50% concession on courses for PWD and Transgender aspirants
Differential Equations: recurrence formulae for Jn(x) – generating function for Jn(x) Hermite differential equation Hermite’s polynomials – Generating function of Hermite polynomials Recurrence formulae for Hermite polynomials – Rodrigue’s formula
Complex variables: analytic function – C-R differential equations – C-R equations in polar form –Laplace’s equation – examples – Cauchy’s integral Theorem and formula – Taylor’s series – Laurent’s series – Singularities of an analysis function – Residues and their evaluation – Cauchy residue theorem – Evaluation of definite integrals (trigonometric functions of cos θ and sin θ only)
Group theory : concept of a group – Abelian group – Generators of finite group – Cyclic groups Group multiplication table – Rearrangement theorem – Sub groups – Lagrange’s theorem for finite group conjugate elements and classes – Group of symmetry of an equilateral triangle Group of symmetry of square – Representation of a group – Reducible and irreducible representation – Schur’s lemmas – Orthogonality theorem – Tensor, beta and gamma
functions: scalars, Contravariant and covariant vectors – Tensors of higher rank – Algebraic operation of tensors – Mixed tensor – Symmetric and anti-symmetric tensors – Quotient law – Beta and Gamma functions : Definitions – Symmetry property of Beta function – Other forms of Beta function – Evaluation of Gamma function – Other forms of Gamma function – Relation between Beta and Gamma functions – Examples.
Lagrangian formulation: Generalized coordinates – Mechanics of a particle and system of particles (momentum and energy) D’Alemberts principle – Lagrange’s equations – Applications (linear harmonic oscillator, simple pendulum isotropic oscillator and electrical circuit) Hamilton’s equations – Applications (simple pendulum, compound pendulum and 20 harmonic oscillator) – Deduction of Hamilton’s principle – Hamilton’s variational principle – Principle of Least action. Canonical transformations : Equation of canonical transformations – Infinitesimal contact transformations – Lagrange and Poisson brackets as Canonical invariants – Equations of motion in Poisson bracket form – Jacobi’s identity – Relation between Lagrange and Poisson brackets – Action angle variables – Euler’s angles – Angular velocity of a rigid body – Euler’s equation of motion – Relativity : Einstein’s Mass – Energy relation – Relation between momentum and energy – Four vectors – Four velocity – Energy – Momentum four vectors – Four force Relativistic classification of particles – Relativistic Lagrangian, Hamilltonian function relativistic Lagrangian Hamiltonian of a charged particle in an E.M field.
General Principles of Quantum Mechanics: Wave packet – Time dependent and time independent Schrodinger equation – Linear vector space – Linear operator – Eigen function and Eigen values – Hermitian operator – Postulates of Quantum Mechanics – Simultaneous measurability of observables – General uncertainty relation – Dirac’s notation – Applications : Square well potential with rigid walls and finite walls – Square potential barrier – Alpha emission – Bloch waves in a periodic potential – Kronig – Penny square-well periodic potential Linear harmonic oscillator: Schrodinger method – Operator method – Delta function – Particle moving in a spherically symmetric potential – System of two interacting particles – Rigid rotator Hydrogen atom – Hydrogen orbitals – Angular Momentum : The angular momentum operators Spin vectors for Spin-(1/2) system – Addition of angular momenta – Time independent and dependent Perturbation theory – Basic concepts – Non degenerate energy levels – Anharmonic oscillator: First-order correction – Ground state of Helium – Effect of electric field on the ground state of hydrogen – Transitions to continuum states – Absorption and emission radiation Einstein’s A and B coefficients – Selection rules – Theory of Scattering : Scattering cross- section Scattering by a central potential : partial wave analysis – Significant number of partial waves Scattering by an attractive square – well potential – Breit-Wiger formula – Scattering length Expression for phase shifts – Integral equation – The Born approximation – Scattering by screened Coulomb potential – Validity of Born approximation – Laboratory and centre of mass co-ordinate system
Electrostatics – Electric charge – electric charge density – Coulomb’s law – Electric intensity –Electric potential – Gauss law- Applications – Boundary value problems in electrostatics – Methods of separation variables in Cartesian co-ordinates. Magneto statics – Ampere’s circuital law – Magnetic scalar potential – Magnetic vector potential – Magnetization and Magnetization current – Magnetic intensity – Magnetic susceptibility. Equation of continuity – Displacement current – Maxwell’s equation – Derivations – energy in electromagnetic fields – (poynting’s theorem). Maxwell’s equation in terms of electromagnetic potentials – Concept of gauge-Lorentz gauge. Plane electromagnetic wave and their propagation – Interaction of electromagnetic wave with matter on microscopic scale. Retarded potentials – Radiation from a linear antenna.
Thermodynamics as phenomenological science – Thermodynamic systems – Closed, open, isolated systems – Thermodynamic processes – Adiabatic, isothermal, isochoric, isobaric, isentropic, cyclical and free expansion processes – Reversible, irreversible and Quasi-static processes – Equation of state – Intensive and extensive variables – The P V diagram. Conversion of work into heat and vice-versa – Efficiency – Kelvin-Planck statement of the second law of thermodynamics – Clausius statement of the second law – Carnot cycle – Carnot refrigerator – Carnot’s theorem and corollary. Equation of state of a gas from Avogadro’s law – Ideal gas equation – Specific heat, internal energy and enthalpy of an ideal gas – Entropy change of an ideal gas – Reversible adiabatic process – Reversible isothermal process. Concept of entropy – Entropy of an ideal gas – The T S diagram – Entropy, reversibility and irreversibility. Microstate and Macrostate of macroscopic system, Phase space and Phase space density, Liouville theorem. Canonical ensemble canonical partition function. – Grand canonical ensemble – Density operator, Spin statistics connection, Grand partition function for ideal Bose and Fermi gases, Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann distributions, Application to Black body radiation: Bose theory(a) Debye theory of specific heat(b) Bose-Einstein condensation – Phase transitions.
UNIT 6: Atomic and Molecular Physics
Electromagnetic spectrum – Absorption or Emission of radiation – Line width – Natural line broadening – Doppler broadening – Pressure broadening – Removal of line broadening – X-ray Spectra – Emission and absorption spectra of X-rays. Regular and irregular doublet laws – X-ray satellites – Photoelectron spectroscopy – Ultraviolet photoelectron spectrometers – XPS techniques and Chemical information from photoelectron spectroscopy – Auger electron spectroscopy. Infrared Spectroscopy – Vibrational Energy of a Diatomic molecule – The Diatomic Vibrating Rotator – The Vibrations of Polyatomic molecules – Rotation – Vibration spectra of Polyatomic molecules – Analysis by Infra-red Techniques – IR spectrophotometer Fourier Transform – IR spectrophotometer – Applications – Frank-Condon principle and dissociation energy. Raman Spectroscopy – Theories of Raman scattering – Rotational Raman Spectra – Vibrational Raman Spectra – Mutual Exclusion principle – Raman Spectrometer Polarization of Raman Scattered light – Structural determination from Raman and IR spectroscopy – Near IR – FT- Raman spectroscopy. Laser Spectroscopy – Basic principles: Comparison between conventional light sources and lasers – Saturation – Excitation methods – Detection methods – Laser Wavelength Setting – Doppler Limited Techniques. Nuclear Magnetic Resonance Spectroscopy – Basic principles – Magnetic resonance – Relaxation processes – Pulsed (Fourier Transform) NMR – Wide line NMR spectrometers – Spectra and molecular structure – Chemical shifts – Spin-spin coupling – Integration – Applications. – Principles of Mossbauer spectroscopy – Chemical shifts – Quadrupole splitting and Zeeman splitting. Applications of Mossbauer spectroscopy.
UNIT 7: CONDENSED MATTER PHYSICS
Elements of X-ray Crystallography and defects in solids – Miller indices – Point groups – Space group – Reciprocal lattice – Bragg’s law interpretation – Structure factor – Fcc and Bcc structures – Electron density distribution experimental techniques for crystal structure studies (powder, Laue, rotation crystal method) – Electron and neutron diffraction methods – Point defects – Color centres – Line defects – Edge dislocation – Screw dislocation – Dislocation method. Semiconductors – Intrinsic semiconductor and extrinsic semiconductor – Mobility, drift velocity and conductivity of intrinsic and extrinsic semiconductors – Carrier concentration in intrinsic and extrinsic semiconductors – Band model. Magnetic properties – Magnetic permeability – Theory of diamagnetism – Langevin’s theory of paramagnetism – Weiss theory – Paramagnetic susceptibility of a solid – Calculation of susceptibility – Quantum theory of paramagnetism determination of susceptibility – Para and diamagnetic materials – Ferromagnetism Spontaneous magnetism in ferromagnetism – Curie-Weiss law – Ferromagnetic domains – domain theory Antiferromagnetism – Structure of ferrites Dielectric properties – Microscopic concepts of polarization – Langevin’s theory of polarization in polar dielectrics – Local fields in liquids and solids – Evaluation of local fields for cubic structure – Clausius – Mossotti relation – Lorentz formula – Ferroelectricity – Dipole theory of ferroelectricity – Classification of ferroelectric materials – Antiferroelectricity – Piezoelectricity – The complex dielectric constant and dielectric loss.
UNIT 8: NUCLEAR AND P ARTICLE PHYSICS
Elements of nuclear Structure and Systematics : Theories of nuclear composition (proton-electron theory, proton-neutron theory) – Mass spectroscopy – Bainbridge and Jordan mass spectrograph – Nier’s mass spectrometer – Deuteron – Magnetic and quadra pole moment of deuteron – Ground state of deuteron – Excited state of deuteron – The meson theory of nuclear force – Yukawa potential – Properties of Stable Nuclei and Nuclei models – Semi empirical mass formula – Nuclear models – Shell models –Magic numbers – Single particle model – Collective model – liquid drop model – Magnetic moments and shell model – Prediction of angular momenta of nuclear ground state – Nuclear Reaction Studies. Conservation laws for nuclear reactions – Nuclear energy – Photo nuclear reaction – fission process – cross sections – Bohr Wheeler theory – Elementary Particles – Classification of elementary particles – Fundamental interactions – Electromagnetic, strong, weak gravitational interactions – Parameters of elementary particles – Conservation laws – Quarks theory.
UNIT 9: ELECTRONICS
Semiconductor Diodes: Operation, characteristics and applications of Zener and Avalanche, Varactor, Schottky – barrier, Tunnel diodes; Construction, operation and Characteristics of BJT, FET and MOSFET-FET amplifier – Negative Resistance and Devices -Uni-Junction transistor and its characteristics – UJT relaxation oscillator – UJT applications – Tunnel diode characteristics and applications – Gunn Diode mechanism – Characteristics and applications SCR – characteristics and applications. IC-Fabrication Technology – Monolithic IC process refining and growth of silicon crystals – Silicon Wafer – Operational Amplifier – Characteristics of ideal and practical Op Amps – Parameters of Op Amp – Theory of inverting amplifier – virtual ground – Theory of non-inverting amplifier – Sinusoidal oscillators – Phase shift oscillator – Wein Bridge oscillator – Crystal oscillator – Multi vibrator – Comparator – Schmitt trigger – Square wave and triangular wave generators – Active filters – Digital Electronics Fundamentals – Number systems – Binary arithmetic – 8421 code-excess – grey code –ASCII code – Logic gates and logic circuits – Boolean algebra – De Morgan’s theorems – Arithmetic circuits – Simplification using Karnaugh’s map – problems.
UNIT 10: EXPERIMENTAL PHYSICS
Measurement of energy and time using electronic signals from the detectors and associated instrumentation – Signal processing – A/D conversion – Multichannel analyzers – Time-of-flight Technique – Coincidence Measurements – True to chance ratio – Correlation studies. Error Analysis and Hypothesis testing – Propagation of errors – Plotting of Graph – Distributions – Least squares fitting – Criteria for goodness of fits – Chi square test – Measurement of fundamental constants : e,h,c – Measurement of high and low resistances, inductance and capacitance – Detection of X-rays, Gamma rays, charged particles, neutrons – Ionization chamber – Proportional counter – GM counter – Scintillation detectors – Solid State detectors – Vacuum Techniques – Basic idea of conductance, pumping speed – Pumps : Mechanical Pump – Diffusion pump – Gauges – Thermocouple gauge – Penning gauge – Pirani gauge – Hot Cathode gauge – Low temperature systems – Cooling a sample over a range up to 4 K – Measurement of low temperatures.
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