METHODS OF COMPUTATIONAL PHYSICS

Syllabus

• Monte Carlo (MC) simulation of spins--Ising model
• Numerical vs. MC integration: Simpson's rule, Gaussian quadrature (orthogonal functions--recursive function evaluation, generating function)
• Probability: Importance sampling, Markov chain, Metropolis algorithm
• Random number generation (RNG)
• Statistics: Variance, standard deviation, standard deviation of the MC mean
• Cluster analysis: Graphs, search, stack
• MC simulation of stock price--geometric Brownian motion
• Random walk: Einstein's law, central-limit theorem
• Random variable: Black-Scholes analysis
• Coordinate transformation: Jacobian, Box-Muller algorithm for RNG of normal distribution
• Interpolation: Least square fit of data
• Quantum MC and kinetic MC simulations
• Molecular dynamics (MD) simulation of particles--Newton's second law of motion
• Numerical differentiation
• Ordinary differential equation (ODE): Symplectic integrators
• Minimization of functions: Conjugate gradient method
• Hybrid MD/MC simulation
• Quantum dynamics simulation of an electron--time-dependent Schrodinger equation
• Partial differential equation (PDE)
• Fourier analysis: Spectral analysis, fast Fourier transform (FFT)
• Electronic structures of molecules--quantum mechanical eigenvalue problem
• Linear algebra: Matrix, orthogonal transformation, rank, singular value decomposition, Krylov subspace
• Matrix eigensystems: Housholder transformation, QL decomposition
• Root finding: Newton-Raphson method