Graduate student curriculum

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