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Graduate student curriculum

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