Waves and Optics

The wave equation with examples from mechanics, optics, electromagnetism and quantum mechanicsThe Definition of a propagating wave in one, two and three dimensionsStanding waves in one, two and three dimensionsThe polarization of a waveThe wave equation in one, two and three dimensionsThe polarization of a wave revisitedBasics of fluid dynamicsWaves in a fluid-derivation from first principlesLongitudinal sound/pressure waves in a tubeThe difference between transverse and longitudinal waves in terms of wave polarizationMaxwell’s equations and the wave equation for the electric and magnetic fields in free spaceSolution to Maxwell’s equations in terms of retarded potentials satisfying the wave equation with sourceThe principle of superpositionDiffraction and interference of wavesGreen’s function for wave equation with sources-Fraunhoffer and Fresnel’s diffractionThe basic Eikonal equation of geometric opticsDescribing the trajectory of light in a medium having spatially varying refractive indexPropagation of light in anisotropic, inhomogeneous and time varying mediumThe Schrodinger wave equation in quantum mechanicsThe effect of noise on the Schrodinger wave equation-Open systems, ie, coupling of the system to the bath environmentWave equation with random non-uniform refractive indexThe relationship between the wave equation and the Helmholtz equation for waves of given frequencyWaves in a confined regionSchrodinger’s wave equation for mixed states in the position kernel domainWaves in general relativity, quantum gravity, plasma physics and quantum stochasticsGravitational wavesQuantum gravity, the canonical ADM formalism-Schrodinger’s equation for the wave function of the space-time metricPlasma wavesEvans-Hudson diffusion as a quantum mechanical generalization of the wave equation with noiseWaves in an expanding universe-Newtonian theory of small fluctuations and general relativistic theory of small fluctuationsEM waves in a curved space-time geometry with inhomogeneous permittivity-permeability tensorQuantum Optics. Here, the photon field is a quantum electromagnetic field expressible as a superposition of annihilation and creation operators of the photon field with the coefficients of the linear combination being positions of time and spaceQuantum optics, notion of a generalized measurement, state collapse after quantum measurement, recovery of states passed through a noisy quantum system, the Knill-Laflamme theorem Stinespring’s representation of noisy quantum systems, Information, relative entropy, mutual information and Renyi entropy of quantum systems. Transmission of information over quantum system. The relevance of all this to the wave mechanics of SchrodingerControlling the quantum em field produced by electrons and positrons by using a classical em field-An application of Dirac’s relativistic wave equationCalculating the path of a light ray in a static gravitational fieldA study of thermal emission by blackholes via Hawking radiation, quantum mechanics of fields in the vicinity of a blackhole and the interaction of electrons,positrons, photons and gravitons with an external noisy bath with application to the design of very large size quantum gatesAnalysis of waves in engineering and optical systems, in biological systems,classical and in quantum blackhole physicsWave digital filter designLarge deviation principle in wave-motionSome more problems in Schrodinger-wave mechanics and Heisenberg-matrix mechanics with relevance to quantum information theoryQuestions in optimization techniquesQuantum antennas via the Schrodinger wave equationLinear algebra for quantum information theoryTransmission lines and waveguides-QuestionsSome more matrix inequalities related to quantum information theoryFresnel and Fraunhoffer diffractionSurface tension and wave propagationKlein-Gordon equation in the Schwarzchild space-time with a radial time independent electromagnetic field and its application to computing the Hawking temperature at which massless/massive particles are emitted from a blackholeQuantum Belavkin filtering versus classical Kushner-Kallianpur filtering-A comparisonRemark on quantum Belavkin filtering for estimating the state of a quantum vibrating stringElementary problems in robotics based on damped simple harmonic motionApproximate solution to the Dirac equation in curved space-timeSome applications of quantum gate design using physical systemsConvergence of perturbation series for nonlinear differential equationsPoiseulle’s law and generalized Poiseulle’s law for flow through a pipeMeasurement of refractive indexModes of a vibrating string with applications to particle physicsHidden Markov Models for estimating the amplitude, frequency and phase of a sinusoidal signal making transitionsThe energy-momentum tensor of the Dirac field in a background curved space-time metricRemark on Noether’s theorem on conserved currentsEnergy-momentum tensor using the tetrad formalismAnalysis of gravitational waves produced by a finite system of point particles-A perturbation theoretic approachHeat equation and its solution in Rn, relationship between heat and wave equations, nonlinear heat equations arising as the scaling limit of the simple exclusion processStudy of wave motion of the boundary of single cellular micro-organsims by giving them external stimulus and observing the wave like motion of their boundary walls as well as wave-like fluctuations of the velocity field of the cytoplasmic fluid within themSnell’s laws of reflection and refraction on surfaces separating two uniform mediaSpinor form of some equations of mathe matical physics:Roger Penrose’s theoryPrisms, mirrors and lenses, the general theoryA brief summary of the bookProbability Theory and Statistics required for random wave motion analysisSummary of ContentsProbability spaces and measure theoretic theorems on probability spacesBasic facts about quantum probabilitySome basic classical and quantum stochastic processesSome applications of classical probability to engineering systemsQuantum stochastic differential equationsSome practical applications of quantum probabilityComputation of scattering cross sectionsThe wave function of our universeComputation of tunneling probabilities of a quantum particle through a potential barrier with applications to tunneling diodeProbability of induced transitions in a laserEnergy bands in a semiconductorModeling quantum systems coupled to a noisy bathCasting the HP equation in functional derivative formCircuit theory concepts from field theory conceptsGraph theoretic analysis of large linear circuits based on KCL and KVLTwo port network theoryDiode and capacitance circuit modelsClassical device physicsDevice physics using quantum mechanics and quantum electrodynamicsBand theory of a semiconductor by solving Schrodinger’s equationQuantum electrodynamics and quantum field theoryAnalyzing random Gaussian and non-Gaussian noise in circuits using higher order correlations and spectraNoise in nonlinear transistor circuitsDigital electronicsTechniques for analyzing transmission linesBrownian motion, Poisson processes and stochastic differential equations in circuit theoryClassical and quantum random processes in circuit theorySimulation of nonlinear ode’s and pde’s in circuit theory and electromagneticsDerivation of medium properties from basic physical principles involving motion of individual electrons and magnetic moments in external fieldsPartial differential equation methods for analyzing waveguidesCurvilinear coordinate systems and variational methods in engineering electromagneticsPerturbation theoretic methods in solving electromagnetics problemsNumerical methods in antenna, waveguide and cavity resonator theoryLarge deviation theory applied to engineering systemsRobotics based on nonlinear differential equationsQuantization of robot motionFiltering and control of engineering systemsQuantum many body systems applied to Fermi operator fields and superconductivityLie group theory in image processingLie group based roboticsLevy process models for jerk noise in robotic systemsDigital systems, classical and quantum gates, design of counters using shift registers and flip-flopsHMM and some of its applicationsQuantum Image ProcessingIntroduce the design aspects of some gadgets through mini-projects some examplesA simple way to introduce quantum electrodynamicsHow to teach the theory of non-Abelian gauge theories as non-commutative generalizations of electromagnetismHow to introduce astronomy and cosmology to undergraduatesQuantum image processing revisitedThe EKF for arbitrary Markov processes with Gaussian measurement noiseQuantum scattering theory applied to quantum gate designSuperconductivity for two species and interpretation of the gap functionIntroductory quantum information theoryQuantum image processingLie group-Lie algebra approach to robot dynamics with two 3 — D links, each described by three Euler angles and an overall translational vector a(t) є R3Linear algebra and operator theoryApplications of Lie groups and Lie algebras, filtering, field quantization, Numerical methods for quantum mechanical problemsquantum random walkLie group-Lie algebra theoretic coordinate free formulation of the equations of motion of a robot with N 3-D links with the orientation of each link described by an arbitrary element of SO(3) and taking in addition into account a translation of the base pivot of the first linkNumerical methods for computing transition probabilities for photons, gravitons, Klein-Gordon Bosons, Dirac Fermions and non-Abelian matter and gauge particles from inside the critical radius to outside of a Schwarzchild blackhole with quantum gate design applicationsNumerical methods for designing quantum gates based on quantum scattering theory for a Schrodinger projectile interacting with a potentialQuantization of a robot in the Lie-group domain when the robot has N 3-D linksLie group formulation of the single 3-D robot link in the presence of gravitation and external torqueQuantum antennas based on non-Abelian matter and gauge fieldsThe electroweak theoryWavelet based system parameter estimationApplying the EKF to nonlinear circuits involving diodes and transistorsAn introduction to classical and quantum error detecting and correcting codesOrthogonal Latin squares and coding theoryCyclic codesYang-Mills field quantization methodsThe Ginzburg-Landau model for superconductivityTeaching the basics of classical mechanics to school students and first year undergraduatesTeaching Linear algebra and functional analysis to post-graduate students of signal processingVariants of the Kalman filterThe Cq-coding theorem:Proof based on Quantum Renyi entropy and Shannon’s random coding argumentManual for the Digital Signal Processing and Statistical Signal Processing LaboratoriesList of Experiments based on the above theoryLinear Algebra and Probability theory in Signal ProcessingLinear Algebra in Signal ProcessingMATLAB problems in waves and opticsMATLAB exercises on classical and quantum information theoryMATLAB exercises involving convergence analysis of the LMS algorithmMATLAB problems on root space decomposition of a Lie algebraCartan’s criterion for semisimplicity of a Lie algebraProblems in linear algebraProblems in non-linear filtering theorySpectral theorem for bounded self-adjoint operators in a Hilbert space-basic stepsMotion of rigid bodies in electromagnetic fieldsLarge deviation theory with engineering applicationsLectures in Linear Algebra for Signal Processing ApplicationsOn the improvement of the signal quality in telephone linesQuantum Coulomb scatteringTutorial problems in electromagnetic field theoryLecture plan for electromagnetic field theory EC-C09A digression into infinite dimensional vector spacesContinuation of finite dimensional vector spacesThe general relativistic Maxwell equations in a resonatorMackey’s theory on the construction of the basic observables in the quantum theory from projective unitary representations of the Galilean groupHamiltonian density of the electromagnetic field in curved space-time in terms of position and momentum fieldsCoulomb scatteringElectromagnetic waves in the Schwarzchild metricTutorial problems in electromagnetic field theoryThe gravitational n-body problem in general relativity: an approximate treatmentLecture plan for ’’Linear Algebra in Signal Processing”-SP-C01Continuation of finite dimensional vector spacesCartan’s equations of structureProof of the Riesz representation theoremQuantum image processingInclusion of the Goldstone boson field in the gauge field after symmetry breakingSome problems in linear algebraGravitational N-body problem in general relativityMultipole radiation fields in the Maxwell theoryHow Dirac brackets are used to take care of constraints in Lagrangian and Hamiltonian mechanicsDeep learning of speech modelsSome problems on linearization (for the course Linear algebra in signal processing)Research project proposal for simulating quantum gates of large sizes using quantized ridge waveguide electromagnetic field interacting with quantum dots and also for estimating medium properties onDesign of a differentiator using series connection of short circuited transmission line elementsDesign of quantum gates by interaction of a quantum em field with gravityScattering theory in the interaction picture for time dependent interactionsChapterwise report on Jaspal Khinda’s Ph.D thesisTraining a DNN with stochastic inputs with analysis of the robustness against input process and weight matrix fluctuationsQuantum Boltzmann equationList of Ph.D scholars supervised by Har-ish Parthasarathy with a brief summary of their thesesThe problem of determining the surface current density induced on an antenna surface placed in a nonlinear inhomogeneous and anisotropic medium taking gravitational effects into account
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