This repository contains Python implementations of advanced numerical algorithms applied to problems in classical mechanics, quantum mechanics, statistical physics, and data analysis. The solutions demonstrate high-performance computing techniques, including Monte Carlo simulations, spectral methods, and sparse matrix operations.

Dependencies

• Python 3.x
• NumPy (Vectorized numerical operations)
• SciPy (ODE solvers, sparse linear algebra, FFT)
• Matplotlib (Data visualization and animation)

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Project Structure

Problem Set 0: Wave Physics & Stochastic Processes

Focus: Visualization and Random Walks

• Interference Patterns: Simulation of scalar monochromatic waves from multiple point sources. Implements 2D intensity heatmaps and time-dependent wave animations to visualize constructive and destructive interference.
• The Gambler's Ruin: Monte Carlo simulation of a stochastic random walk (1D). Analyzes statistical properties such as the probability of ruin and the mean time to ruin, verifying theoretical predictions for "fair" and "unfair" games.

Problem Set 1: Linear Algebra & 1D Quantum Systems

Focus: Algorithm Complexity and Eigenvalue Problems

• Gaussian Elimination: Implementation of a linear solver from scratch, including row reduction and partial pivoting. Includes performance profiling (log-log analysis) to demonstrate $O(N^3)$ scaling compared to SciPy's optimized routines.
• 1D Tight-Binding Models: Construction of Hamiltonians for discrete 1D quantum chains.
  • [cite_start]Random Chain: analysis of Anderson localization-like effects.
  • [cite_start]Su-Schrieffer-Heeger (SSH) Model: Study of topological phases and edge states in a dimerized chain.

Problem Set 2: Dynamical Systems & Time-Evolution

Focus: ODE Integration and Spectral Methods

• Three-Body Problem: Numerical solution of coupled differential equations for gravitational N-body systems using scipy.integrate.ode. Includes visualization of chaotic orbits and Fourier analysis (FFT) of trajectories to identify periodicity.
• Split-Step Fourier Method: Solver for the time-dependent Schrödinger equation (TDSE) in 1D and 2D. Utilizes Fast Fourier Transforms (FFT) to alternate between position and momentum space for efficient time-stepping.Features animations of wavepackets in harmonic and circular well potentials.

Problem Set 3: 2D Quantum Statics

Focus: Sparse Matrices and Finite Difference Methods

• 2D Time-Independent Schrödinger Equation: Finite-difference eigensolver for arbitrary 2D potentials (e.g., "Particle in a Box", "Stadium Billiard"). Utilizes scipy.sparse and Arnoldi iteration (eigsh) to efficiently solve for low-lying energy states.
• Honeycomb Lattice Tight-Binding: Simulation of complex 2D quantum lattices (Graphene-like structures).Implements magnetic vector potentials to study the effects of external magnetic fields on energy spectra.

Problem Set 4: Statistical Mechanics & Bayesian Inference

Focus: Markov Chain Monte Carlo (MCMC)

• 2D Ising Model: Metropolis Monte Carlo simulation of spin systems on a square lattice.Calculates thermodynamic quantities (Magnetization, Heat Capacity, Susceptibility) and implements re-weighting techniques to estimate properties across temperatures from a single run.
• Bayesian Parameter Estimation ($\Lambda$-CDM): Application of MCMC to fit cosmological models to observational Hubble data.Estimates posterior distributions for the Hubble constant ($H_0$), matter density ($\Omega_m$), and dark energy density ($\Omega_{\Lambda}$).

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Usage

Each problem is contained in a separate Python script. To run a specific simulation, execute the file from the terminal:

python ps0_interference.py