Errors and uncertainties in computations: Types of errors, error in functions, errors in algorithms. Matrix computing and scientific libraries.
Zero-finding and matching: Newton's rule for finding roots. Quantum eigenvalues, particle in a box. Fields due to moving charges.
Integration: Trapezoid rule, Simpson's rule, Gaussian quadrature, multi-dimensional integrals. Monte-Carlo integrations.
Phase Transitions and Critical Phenomena
General Introduction: Origin of phase transition, thermodynamic instabilites, Maxwell construction. Classification of phase transitions: first order and second order. Phase transitions in different systems (e.g. liquid-gas and paramagnet-ferromagnetic transition), order parameter, critical exponents, concept of long-range order.
Lattice models: Ising Model, exact solution in one dimension, high-temperature and low-temperature expansions. Phase transitions in X-Y and Heisenberg Models.
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• Numerical modeling on point spread function of perfect and aberrated systems
• Numerical modeling on focusing by lens let arrays
• Numerical modeling on the image formation by perfect/and aberracted systems
• Numerical modeling of Zernike polynomials of the aberrated wavefront
• Experiment with wavefront sensor:
◦ Measurement of aberrated and un-aberrated wavefront
◦ Corrections of aberrated wavefronts
◦ Evaluation of Zernike polynomials