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