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

#### 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
• 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 coefficientsHow/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

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