Graduate student curriculum

Owned by Christian Santangelo
Mar 21, 2023

This page contains a minimal curriculum for the first-year graduate classes. The curriculum is still being updated and developed.

Fall semester

Classical Mechanics

Pre-requisites: Basic calculus including changes of variables, ordinary differential equations, basic calculus of variations at the undergraduate level

  • Newtonian mechanics in arbitrary coordinates
  • Lagrangian mechanisms and Hamilton's principle
  • Mechanical systems with constraints, Lagrange multipliers
  • Symmetries and conservation laws; Noether's theorem
  • Central force motion, gravitational 2-body problem
  • Small oscillations
  • Rigid body motion, moment of inertia tensor
  • Euler's equations
  • Hamiltonian dynamics, Hamilton's equations

Quantum 1

Pre-requisites: Basic familiarity with the time-independent Schrodinger's equation in 1D and 3D, expectation values, undergraduate understanding of boundary value problems

  • Mathematical formalism of quantum mechanics
  • Postulates of quantum mechanics
  • Uncertainty relations
  • Spin 1/2 precession
  • 1D wave mechanics
  • Simple harmonic oscillator and coherent states
  • Rotations in 3D
  • Angular momentum
  • Orbital angular momentum and spherical harmonics
  • Hydrogen atom

Spring semester

Electricity and Magnetism

Pre-requisites: Electrostatics at an advanced undergraduate level including Laplace's equation and boundary value problems

  • Basics of electrostatics (electric fields and electric potentials) 
  • Conductors and charged points, lines, and surfaces 
  • Electrostatic energy 
  • Multipole expansion (electrostatics) 
  • Laplace’s equation 
  • boundary value problems in different 3D coordinate systems (for spherical harmonics – note overlap with Quantum I here) 
  • Dielectrics, energy in dielectric 
  • Magnetostatics, current, surface current, Ampere’s law 
  • Vector potential, gauge transformations 
  • Multipole expansion (magnetostatics) 
  • Maxwell’s equations, gauge transformations, wave equations for light in vacuum 
  • Plane waves, polarization in vacuum and matter 
  • Refraction, reflection, diffraction 
  • Electromagnetic radiation and scattering 

Quantum 2

Pre-requisites: Solutions of Schrodinger equation: Free particle, Particle in a box, 1D Harmonic oscillator, Familiarity with hydrogen atom, Some angular momentum 

  • Discrete symmetries 
  • Many particles; exchange symmetry; Fermions and bosons (suggest this appear in 1st half of semester to complement Stat. Mech.) 
  • Unitary and anti-unitary operators, continuous symmetries, and generators 
  • Angular momentum and spin 
  • Addition of angular momentum 
  • Wigner-Eckhardt theorem and tensor operators 
  • Time-independent perturbation theory (degenerate and non-degenerate) 
  • Fine structure of the hydrogen atom 
  • Time-dependent perturbation theory 
  • Fermi’s Golden rule and applications 

Statistical Mechanics

Pre-requisites: Binomial coefficients, How/where/when to apply Taylor series and approximations , Basic probability operations and how to stack multiple and/or possibilities (coin flips, dice rolls, etc.), Differentiation and integration, volume and surface integerals, minima & maxima, infinite sums 

  • 1st law of thermodynamics 
  • Entropy and the 2nd law of thermodynamics 
  • Thermodynamic potentials, 3rd law of thermodynamics 
  • Chemical potential 
  • Microcanonical ensemble 
  • Canonical ensemble, Grand canonical 
  • Statistical ensembles and examples: harmonic oscillator, paramagnetism, diatomic gas, etc. 
  • Ideal gas 
  • Equations of state, van der Waal’s gas  
  • Phase transitions 
  • Quantum statistics: Symmetrized wave functions, Bose and Fermi statistics 
  • Density of states 
  • Examples: one or more of Free fermi gas, Bose condensate