Theoretical Physics Course

Lesson 1 • Ordinary and Partial Differential Equations

Solves ODEs and PDEs central to wave, heat, and Schrödinger equations. Connects solution methods to boundary and initial value problems in physics.

Lesson 2 • Probability, Statistics, and Fourier Methods

Covers Fourier series, transforms, and probability distributions used in physics. Provides tools for spectral analysis and statistical mechanics.

Lesson 3 • Complex Analysis and Contour Integration

Introduces analytic functions, Cauchy's theorem, and residue calculus. These tools are indispensable for solving integrals in quantum and statistical physics.

Lesson 4 • Linear Algebra for Physicists

Develops matrix theory, eigenvalue problems, and vector spaces. These structures underpin quantum mechanics, classical mechanics, and field theory.

Lesson 1 • Lagrangian Mechanics

Derives equations of motion from the principle of least action using generalised coordinates. Demonstrates how symmetry yields conservation laws via Noether's theorem.

Lesson 2 • Hamiltonian Mechanics

Introduces the Hamiltonian, phase space, and Hamilton's equations. Lays the groundwork for canonical quantisation and statistical mechanics.

Lesson 3 • Newtonian Mechanics Revisited

Reviews Newton's laws, constraints, and conservation principles with precision. Establishes the conceptual baseline before introducing advanced formulations.

Lesson 4 • Hamilton-Jacobi Theory and Action-Angle Variables

Solves mechanics problems via the Hamilton-Jacobi equation and action-angle variables. Bridges classical integrability to semiclassical quantum mechanics.

Lesson 1 • Maxwell's Equations and Electromagnetic Waves

Derives Maxwell's equations from physical laws and shows they predict electromagnetic waves. Establishes the field-theoretic structure central to modern physics.

Lesson 2 • Covariant Formulation of Electrodynamics

Rewrites Maxwell's equations using the electromagnetic field tensor. Demonstrates manifest Lorentz covariance and derives the stress-energy tensor.

Lesson 3 • Relativistic Dynamics and Four-Vectors

Extends mechanics to relativistic regimes using four-vectors and the energy-momentum relation. Introduces the covariant formulation of physical laws.

Lesson 4 • Special Relativity: Kinematics

Establishes the postulates of special relativity and derives Lorentz transformations. Analyses time dilation, length contraction, and relativistic velocity addition.

Lesson 1 • Exactly Solvable Quantum Systems

Solves the infinite square well, harmonic oscillator, and hydrogen atom exactly. Builds physical intuition for quantisation, energy levels, and wave functions.

Lesson 2 • Operators, Observables, and Measurement

Develops Hermitian operators, eigenvalue spectra, and the measurement postulate. Derives the uncertainty principle from commutation relations.

Lesson 3 • Postulates and Wave Functions

States the postulates of quantum mechanics and interprets the wave function probabilistically. Introduces the Schrödinger equation as the fundamental dynamical law.

Lesson 4 • Quantum Dynamics and Pictures

Contrasts Schrödinger, Heisenberg, and interaction pictures of time evolution. Derives the time evolution operator and applies it to two-level systems.

Lesson 1 • Identical Particles and Many-Body Systems

Introduces symmetrisation postulate, Slater determinants, and second quantisation. Connects particle statistics to Fermi-Dirac and Bose-Einstein distributions.

Lesson 2 • Relativistic Quantum Mechanics

Derives the Klein-Gordon and Dirac equations as relativistic wave equations. Introduces antiparticles, spin from the Dirac equation, and the non-relativistic limit.

Lesson 3 • Angular Momentum and Spin

Constructs the full angular momentum algebra and introduces intrinsic spin. Derives Clebsch-Gordan coefficients for addition of angular momenta.

Lesson 4 • Scattering Theory

Derives the Born approximation, partial wave expansion, and optical theorem. Connects theoretical cross sections to experimental scattering measurements.

Lesson 1 • Canonical and Grand Canonical Ensembles

Derives partition functions for canonical and grand canonical ensembles. Computes thermodynamic quantities from free energies and chemical potentials.

Lesson 2 • Quantum Statistical Mechanics

Applies statistical mechanics to quantum systems obeying Fermi-Dirac and Bose-Einstein statistics. Derives properties of ideal quantum gases and their physical consequences.

Lesson 3 • Phase Transitions and Critical Phenomena

Classifies phase transitions and introduces order parameters, mean-field theory, and critical exponents. Connects scaling laws to universality classes.

Lesson 4 • Non-Equilibrium Statistical Mechanics

Introduces Boltzmann transport equation, linear response theory, and fluctuation-dissipation theorem. Applies these to transport coefficients and irreversible processes.

Lesson 1 • Classical Field Theory

Applies the Lagrangian formalism to fields, deriving Euler-Lagrange field equations. Introduces Noether's theorem for fields and the stress-energy tensor.

Lesson 2 • Interacting Fields and Feynman Diagrams

Develops perturbation theory for interacting fields using Wick's theorem and Feynman rules. Computes S-matrix elements and physical scattering amplitudes.

Lesson 3 • Path Integral Formulation

Derives the path integral for quantum mechanics and extends it to field theory. Uses functional methods to derive propagators and generate Feynman diagrams.

Lesson 4 • Renormalisation and Regularisation

Identifies and removes ultraviolet divergences using dimensional regularisation and renormalisation. Introduces the renormalisation group and running coupling constants.

Lesson 1 • Differential Geometry for Physicists

Introduces manifolds, tensors, covariant derivatives, and curvature. Provides the geometric language required to formulate general relativity precisely.

Lesson 2 • Equivalence Principle and Geodesics

States the equivalence principle and derives geodesic equations as the relativistic law of free fall. Connects spacetime curvature to gravitational effects.

Lesson 3 • Einstein's Field Equations

Derives Einstein's field equations from the Einstein-Hilbert action. Analyses the stress-energy tensor as the source of spacetime curvature.

Lesson 4 • Gravitational Waves and Cosmology

Derives gravitational wave solutions from linearised Einstein equations and introduces the FLRW metric for cosmology. Connects theory to observational evidence.