# MATHEMATICAL METHODS-I Lecture-1

### MATHEMATICAL METHODS-I

Vector Analysis:
Review of vectors in the Algebra, Vector in thedifferentiation and gradient, Divergence and Gauss’s theorem, Vector integration, in the Green’s theorem in the plane, Curl and Stokes theorem.
Curvilinear Coordinates and Tensors:
Curvilinear coordinate in the system, Gradient, Divergence in the and Curl in the curvilinear in the coordinates system,in the  Cartesian, Spherical and in the Cylindrical coordinate system, Covariant and contra variant tensors, Tensor algebra, in the Quotient rule.
Matrices:
Linear vector spaces,in the  Determinants, Matrices, Eigenvalues andin thein the eigenvectors of matrices, Orthogonal matrices, in the Hermitian matrices, Similarity transformations, in the Diagonalization of matrices.
Group theory:
Introduction to groups, in the Group representation, Invariant subgroups, in the discrete groups-Dihedral groups, Continuous in the groups-O groups, SU (2) groups, Lie groups
Complex Variables:
Functions of a in the complex variable, Cauchy Riemann conditions and in the analytic functions, Cauchy integral theorem and integral formula, in the Taylor and Laurent series, in the Calculus of residue, in the Complex integration.
QUANTUM MECHANICS-I
Arfan bajwa
Quantum Mechanics of One Dimensional Problem:
in the Review of concepts of classical mechanics, in the State of a system, Properties of one dimensional potential functions, Functions andin the expectation values, in the Dirac notation, Hermitian operators, in the Solutions of Schrodinger in the equation for free particles, The potential barrier problems, The linear in the harmonic oscillator, Particle in a box.
Formalism of Quantum Mechanics:
The state of a system, in the Dynamical variables and operators, Commuting and non-commuting operators, Heisenberg in the uncertainty relations, Time in the evolution of a system, in the  Schrodinger and Heisenberg pictures, Symmetry in the principles and conservation laws.
Angular Momentum:
Orbital angularin the momentum, Spin, The eigenvalues and Eigen functions of L2 and Lz, Matrix representation of angular momentum in the operators, Addition of angular in the momenta.
Schrodinger Equation in Three Dimensions:
Separation of Schrodinger in the equation in the Cartesian coordinates, Central potentials, The free particle, Three dimensional square in the well potential, The hydro genic atom, Three dimensional square in the well potential, The hydro genic atom, in the Three dimensional isotopic oscillator.

THERMAL AND STATISTICAL PHYSICS
Equilibrium Thermodynamics:
Basic postulates,in the  fundamental equations and in the equations of in the state, response functions Maxwell’s relation, reduction in the of derivatives.
Elements of Probability Theory:
Probabilities, in the distribution functions, statistical in the interpretation of entropy, in the Boltzmann H-theorem.
Formulation of Statistical Methods:
Ensembles,in the  counting of states (in classical and quantum mechanicalin the  systems, examples) partition function, Boltzmann distribution. Formation ofin the  Microcononical, canonical and grand canonical partion function.
Partition Function:
Relations of partition function in the with thermodynamic in the variables, examples (collection ofin the  simple harmonic oscillators, Pauli and in the Van Vleck paramagnetics, in the Theorem of equipartition of energy.
Statistical Systems:
in the Maxwell-Boltzmann, in the Bose-Einstein, Fermi-in the Dirac statistical systems. Examples in the of thermodynamics of these systems; in the Black body radiations, Gas of electrons in the solids.
Statistical Mechanics of Interacting Systems:
in the Lattice vibrations in solids; Van der in the Waals Gas: mean field calculation; Ferro magnets in Mean Field in the Approximation.
in the Fluctuations, Bose-Einstein in the Condensation, Introduction to in the density matrix approach.

CLASSICAL MECHANICS
Arfan Bajwa
Elementary Principles:
in the Brief Survey of Newtonian in the mechanics of a system of in the particles, constraints, Alembert’s principle, Lagrange’s in the equation, and its applications. Virtual work.
Variational Principles:
Calculus of variation and in the Hamilton’s principle, Derivation of Lagrange’s in the equation from in the Hamilton’s principle.

Two Body Central Force Problems:
Low and least action, two body problem and its reduction to one body problem. Equation of motion and solution for one body problem, Kepler’s Laws Laboratory and Centre of mass systems, Rutherford scattering.
Kinematics of Rigid Body Motion:
in the Orthogonal transformations, in the Eulerian angles, Euler’s theorem, the cariole’s force
Rigid Body Equation of Motion:
Angular momentum, in the Tensors and dyadic, Moment of in the inertia, Rigid body problems and in the Euler’s equations.
Hamilton Equation of Motion:
in the Legendre transformation and in the Hamilton equations of motion, Conservation in the theorems.
Canonical Transformations:
Examples of canoical transformations, in the Lagrange and Poison brackets,in the  Liouville’s theorem.

ELECTRONICS
Irfan butt
Special Diodes:
Zener diodes, in the Zener regulators, Varactor in the diodes, Schottky diodes, Light emitting diodes, Photodiodes, Tunnel diodes, Varistors and their applications.

Transistor Circuits:
Bipolar in the transistors; in the parameters and ratings, Ebers-Moll,in the  Hybrid-p and h,z and y-parameterin the models, Switching in the circuits, in the Biasing and stability, in the Common emitter, Common base and common collector in the amplifiers, Frequency in the response, Power class A, B, and in the C amplifiers, Field Effect
FET
Transistors; in the Junction FET, in the MOSFET, Operation  in the and construction, Biasing, Common source and common drain in the amplifiers, Frequency response. Multistage Amplifiers; RC coupled and direct coupled stages, The differential amplifiers, in the Negative feedback, Tuned in the RF Voltage amplifiers, I-F Amplifiers and automatic gain control.
Operational Amplifiers:
Ideal op-amps, in the Simple op-amp arrangements, its data and sheet in the parameters, Non inverting and inverting circuits, Feedback and stability,in the Op-amp applications; Comparators,in the Summing, Active filters, Integrator and in the Differentiator, Instrumentation in the amplifier.
Oscillators:
Armstrong, in the Hartley, CMOSS, Colpit’s Phasein the  shift and 555 timer in the oscillators.
Voltage Regulators:
Series, Shunt in the and switching regulators, Power in the supply. 