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Practical Question: Python
Numerical Methods Problem Set
Learning Objectives:
- Review all built-in, NumPy, and math functions used across typical numerical methods problems given at the end of this page.
- Understand and apply key numerical methods including root finding, interpolation, curve fitting, numerical integration, and solving ODEs.
- Practice basic numerical algorithms using Python.
Numerical Methods
Eigenvalues and eigenvectors play a central role in linear algebra, with wide applications in physics, engineering, and data science. They help understand the action of a linear transformation in a given vector space.
Interaction of Solids with EM Field
Learning Objectives:
Polarons
In solid-state physics, polarons are quasiparticles formed due to the interaction of an electron (or hole) with the phonons (quantized lattice vibrations) in an ionic crystal. This interaction leads to a modification of the electron’s motion, as it becomes “dressed” with a polarization cloud of lattice distortion.
Polaritons
In solid-state physics, polaritons are quasiparticles arising from the strong coupling of photons with optical phonons in a crystal. These coupled modes play a central role in understanding the optical properties of ionic crystals, particularly in the infrared frequency range.
Tight-Binding Approximation
Nearly Free Electron Model and Energy Bands in One Dimension, Tight-Binding Approximation
Small Oscillations
Small Oscillations, Normal Modes of Vibration, Coupled Oscillators
Poisson Bracket, Poisson Theorems
Learning Objectives:
Plasma Oscillations and Plasmons
Learning Objectives:
Hamilton–Jacobi Equation
Hamilton–Jacobi Equation with Example of Harmonic Oscillator
Generating Function
Learning Objectives:
Legendre Transformation
Learning Objectives:
Hamilton Equation of Motion
Hamilton’s Equations of Motion
Hamilton’s Principle
Hamilton’s Principle
Least Action Principle
The Principle of Least Action
Calculus of variation
Calculus of variation
Lagrange’s Equation
D’Alembert’s Principle, Lagrange’s Equation and Its Simple Applications
Dielectric Properties of Materials
Macroscopic Dielectric Constant
Image Processing: Eigenvalues and Eigenvectors
🧠 Objective
This lecture explores the application of eigenvalues and eigenvectors in image processing using Principal Component Analysis (PCA). We will: