Graduate Program Course Outlines
The Course Outlines and syllabi on this web page represent the best descriptions of some of the graduate courses that are available at this time. The Teaching Activities Committee of the Physics Department expects that these topics be covered, but some deviations may result. You should consult the instructor before choosing to take a course because of a specific topic.
Courses:
 Physics 316  Advanced Classical Mechanics (Autumn)
 Physics 322  Advanced Electrodynamics And Optics I (Winter)
 Physics 323  Advanced Electrodynamics And Optics II (Spring)
 Physics 330  Mathematical Methods of Physics (Autumn)
 Physics 341, 342  Quantum Mechanics I and II (AutumnWinter)
 Physics 352  Statistical Mechanics (Spring)
 Physics 353  Advanced Statistical Mechanics (Autumn)
 Physics 361  Solid State Physics (Autumn)
 Physics 363  Introductory Particle Physics (Spring)
 Physics 364  Introduction to General Relativity (Winter)
 Physics 366  Advanced Solid State Physics/Hard Condensed Matter (Winter)
 Physics 367  Soft Condensed Matter Physics (Winter)
 Physics 385  Advanced Mathematical Methods of Physics (Winter/Spring)
 Physics 386  Advanced Methods of Data Analysis (Winter/Spring)
 Physics 443, 444, 445  Quantum Field Theory (AutumnSpring)
PHYS 316: Advanced Classical Mechanics
Level: “Classical Mechanics” by Landau & Lifshitz or “Classical Mechanics” by Goldstein, Poole, and Safko.

Variational Principles, Lagrangians, and Hamiltonians
 Hamilton's principle and Lagrangians
 Hamiltonians
 Other variational principles
 Symmetries and conservation laws

Some Applications
 Central force problems
 Rigid body motion
 Small oscillations

Continuous Systems and Fields
 Continuum mechanics as a limit of particle systems
 Lagrangian and Hamiltonian formulations

General Hamiltonian Formulation
 Poisson brackets
 Canonical transformations
 HamiltonJacobi theory
 Actionangle variables
 Adiabatic invariants
 Chaos in Hamiltonian systems
PHYS 322: Advanced Electrodynamics and Optics I
Level: "Classical Electrodynamics" by Jackson

Electrostatics
 Gauss' law, potentials
 Poisson equation, boundary conditions, methods for solving boundary value problems
 Eigenfunction & multipole expansions
 Conformal mappings of potentials (optional)

Maxwell's Equations
 Potentials, Gauge Transformations
 Conservation laws, Poynting vector, Stressenergy tensor
 Wave equation

Wave Propagation
 Polarization
 Dispersive materials, phase/group velocities
 Propagation in plasmas, conductors, dielectrics
 Waves at media interfaces, Fresnel equations
 Waveguides (optional)
 Special materials, negative index (optional)

Optics and Diffraction
 Diffraction: Fraunhofer and Fresnel
 Scattering of plane waves by a conducting sphere
 Fourier optics, filtering (see "Intro to Fourier Optics" by Goodman)
 Gaussian beams, transfer matrix (optional but recommended)
PHYS 323: Advanced Electrodynamics and Optics II
Level: "Classical Theory of Fields" by Landau and Lifschitz

Review of Special Relativity; covariant forms
 Lorentz covariance of Maxwell's equations
 Relativistic action, EM Lagrangian, Hamiltonian
 Charged particle action, Lorentz force

Coupling of continuous matter to fields
 Dielectrics, thermodynamic properties
 Hall effect and thermoelectric phenomena (optional)
 LandauGinzburg superconductivity (optional)

Radiation
 Covariant Green's function, retarded time
 Retarded potentials
 Multipole radiation
 Point currents, scattering plane waves from point charges
 Bremsstrahlung, synchrotron, Cherenkov radiation

Special Topics (at instructor's discretion)
 Vortices, topological defects in fields/matter
 Metamaterials
 Physics of charged particle beams, optics of beams
 Topological defects
 Symmetries of the EM field tensor under dilatation, Noether currents
 Electromagnetism and differential geometry
Physics 330: Mathematical Methods of Physics
Level: Matthews and Walker, Mathematical Methods of Physics or Arfken and Weber, Mathematical Methods for Physicists

Linear Algebra
 Finitedimensional vector spaces
 Functional analysis

Complex Analysis
 Analytic functions
 Contour integration

Ordinary Differential Equations
 Exact solutions
 Series solutions
 Approximation methods

Partial Differential Equations
 Hyperbolic, elliptic and parabolic equations
 Kernels
Note: This outline is intended as a guide to the most essential topics for this course; there is some flexibility in the order and manner of presentation. In particular, examples of particular applications are left to the instructor.
Physics 341342: Quantum Mechanics III
Level:
R. Shankar, Principles of Quantum Mechanics
J. Sakurai, Introduction to Quantum Mechanics

Fundamentals of Hilbert Space
 Vector spaces and Hilbert spaces
 Dirac notation
 Selfadjoint and Unitary operators and their spectra
 Symmetries and unitary transformations
 Projection operators

Simple Quantum Systems and the Relation to Classical Mechanics
 Structure of QM
 Uncertainty Relations
 Two state systems
 Onedimensional problems
 Coherent States
 The classical limit of QM

Time Evolution
 Time evolution operator
 Heisenberg equations of motion
 Heisnberg vs. Schrodinger representation

Symmetry in Quantum Mechanics
 Angular Momentum and commutation relations
 Spin
 Addition of angular momentum
 WignerEckart theorem
 Identical particle and spinstatistics

Interaction with electromagentic fields
 Gauge invariance
 AharonovBohm effect
 Magnetic monopoles
 Stark effect
 Landau levels
 Quantum Hall effect

Perturbation Methods
 Stationary perturbation theory and applications
 Time dependent perturbation theory
 Fermi's golden rule
 Emission and absorption of radiation

Scattering Theory
 General formulation for scattering
 Cross sections and the scattering amplitude
 Definition of Smatrix and analytic properties
 Scattering of identical particles

Path Integral Methods
 Defining sums over paths
 Relation to standard formalism
 Phase space path integrals
 Evaulating gaussian integrals
 Tunneling via instantons

Adiabatic approximation and Berry's phase
 BornOppenheimer or adiabatic approximation
 Sudden approximation
 Berry's phase and potential
 Global interpretation, examples

Measurement theory and decoherence
 Bell's inequalities
 Schrodinger's cat and the problem of collapse of the wave function
 Decoherence, basic idea and simple models

Supersymmetric quantum mechanics
 Supersymmetry
 Supersymmetry in quantum mechanics
 Solvable examples
 Supersymmetry and index theory
Note: Topics 1  7 should be considered core material that should be covered every year. There should usually be time to cover some of the remaining four topics (or other topics chosen by the instructor).
PHYSICS 352: STATISTICAL MECHANICS
Level: "Statistical Physics of Particles" by Kardar

Thermodynamics
 The first and second laws
 Carnot engines
 Entropy
 Approach to equilibrium and thermodynamic potentials
 Maxwell relations and stability conditions
 The third law

Probability
 One random variable
 Some important probability distributions
 Many random variables
 Sums of random variables and the central limit theorem
 Rules for large numbers
 Information, entropy, and estimation

Kinetic Theory of Gases
 Liouville's theorem
 The BogoliubovBornGreenKirkwoodYvon hierarchy
 The Boltzmann equation
 The Htheorem and irreversibility
 Equilibrium properties
 Conservation laws
 Hydrodynamic limit

Classical Statistical Mechanics
 The microcanonical ensemble
 The ideal gas
 The canonical ensemble
 The Gibbs canonical ensemble
 The grand canonical ensemble
 Examples

Interacting Particles (optional)
 The cumulant expansion
 The cluster expansion
 The second virial coefficient and van der Waals equation
 Meanfield theory of condensation
 Variational methods
 Corresponding states
 Critical behavior

Quantum Statistical Mechanics
 Dilute polyatomic gases
 Vibrations of a solid
 Blackbody radiation
 Quantum microstates
 Quantum macrostates

Ideal Quantum Gases
 Hilbert space of identical particles
 Canonical formulation
 Grand canonical formulation
 Nonrelativistic gas
 The degenerate Fermi gas
 The degenerate Bose gas
 Superfluid He4
PHYSICS 353: ADVANCED STATISTICAL MECHANICS
Level:
"Statistical Physics of Fields" by Kardar
"Scaling and Renormalization in Statistical Physics" by Cardy

Collective Behavior, From Particles to Fields
 Phonons and elasticity
 Phase transitions in simple systems
 Critical behavior

Mean Field Theory
 Coarse graining and the LandauGinzburg Hamiltonian
 Saddle point approximation, mean field theory
 Continuous symmetry breaking and Goldstone modes
 Discrete symmetry breaking and domain walls

Fluctuations
 Scattering and fluctuations
 Correlation functions and susceptibilities
 Lower critical dimension
 Fluctuation corrections to the saddle point
 The Ginzburg criterion

The Scaling Hypothesis
 Divergence of the correlation length
 Critical correlation functions and selfsimilarity
 The renormalization group idea
 Block spins
 Onedimensional Ising model
 Scaling behavior of the free energy
 Critical exponents
 Scaling for the correlation functions
 Scaling operators and scaling dimensions
 The Gaussian model

Perturbative Renormalization Group
 Diagrammatic representation of perturbation theory
 Susceptibility
 Perturbative renormalization group
 The epsilonexpansion
 The Gaussian fixed point, the WilsonFisher fixed point
 Logarithmic corrections in d = 4
 The O(n) model near four dimensions

Lattice Systems (optional)
 Models and methods
 Transfer matricies
 Position space renormalization group in one dimension
 The MigdalKadanoff bond moving approximation

Series Expansions
 Lowtemperature expansions
 Hightemperature expansions
 Exact solutions of the 1dimensional Ising model
 Selfduality in the 2dimensional Ising model
 Exact free energy of the square lattice Ising model
 Critical behavior of the 2dimensional Ising model

Lowdimensional Systems and Topological Defects
 The nonlinear sigma model
 Topological defects in the XY model
 Renormalization group for the Coulomb gas
 Twodimensional solids
 Twodimensional melting

Dissipative and Critical Dynamics (optional)
 Brownian motion of a particle
 Equilibrium dynamics of a field
 Dynamics of a conserved field
 Nonequilibrium dynamics of open systems
 Dynamics of a growing surface

Random Systems (optional)
 Quenched and annealed disorder
 The Harris criterion
 Perturbative approach to the random fixed point
 Percolation
 Random fields

Conformal Symmetry (optional)
 Conformal transformations
 Simple consequences of conformal symmetry
 The stress tensor
 The ctheorem
PHYSICS 361: SOLID STATE PHYSICS
Level: Ashcroft and Mermin, Solid State Physics

Properties of Insulators
 Crystal Lattice Structures
 XRay Scattering and Reciprocal Lattice
 Ground State Properties

Lattice Vibrations, Harmonic Theory and Phonons
 Thermodynamics (Debye Theory) Spatial
 Structure (DebyeWaller factor)

Electronic Properties of Solids

Electrons in a Fixed Periodic Potential (Band Theory)
 Bloch’s Theorem and Perturbation Theory
 Tight Binding Systems
 Density of States

Thermal Properties
 Insulators and SemiConductors
 Metal
 Optical Properties of Solids
 Transport in Metals (conductivity, Hall effect, etc.)

Electrons in a Fixed Periodic Potential (Band Theory)
Physics 363: Particle Physics
Main Text: Griffiths, Introduction to Elementary Particles
Supplementary Texts:
Halzen and Martin, Quarks and Leptons
Kane, Modern Elementary Particle Physics
Perkins, Introduction to High Energy Physics

Overview
 Observed particles
 Forces

Special Relativity and Classical Field Theory Review
 Spacetime and 4vectors
 Relativistic kinematics
 Field theory: Lagrangians, electromagnetism, gauge invariance

Feynman Diagrams
 Timedependent perturbation theory
 Feynman rules (at tree level)
 Crosssections and decay rates

Symmetries
 Group theory review
 SU(2) isospin, product representations, SU(3)
 C, P, and T

Quantum Electrodynamics
 Spinors and fermions
 Feynman rules
 QED processes, Dirac matrix technology

Hadrons and Partons
 ElectronQuark interactions
 Inelastic scattering, partons
 Structure functions

Quantum Chromodynamics
 YangMills theory
 Quarks; Feynman rules
 Running couplings

Electroweak Theory
 Spontaneous Symmetry Breaking
 Bosonic sector: vector bosons, Higgs
 Fermions, SU(2)
 Generations, CKM matrix, CP violation
Physics 364: General Relativity
Main Text: Wald, General Relativity
Supplementary Texts:
Schutz, A First Course in General Relativity
Weinberg, Gravitation and Cosmology
Misner, Thorne, and Wheeler, Gravitation
D'Inverno, Introducing Einstein's Relativity

Special Relativity
 Lorentz Transformations
 Spacetime Diagrams
 Vectors and Tensors
 Proper Time
 Physics in Flat Spacetime

Manifolds
 Coordinate Systems
 Vectors as Derivatives
 Tensor Transformation Law
 The Metric
 Tensor Densities

Curvature
 Covariant Derivatives and Connection Coefficients
 Parallel Transport
 Geodesics
 The Riemann Tensor
 Geodesic Deviation

Gravitation
 The Principle of Equivalence
 Physics in Curved Spacetime
 Einstein's Equations
 The Newtonian Limit

Weak Fields and Gravitational Radiation
 The WeakField Limit
 Linearized Einstein Equations
 Gravitational Waves

The Schwarzschild Solution and Black Holes
 Birkhoff's Theorem
 Geodesics of Schwarzschild
 Kruskal Extension
 Penrose Diagrams
 Charged and Rotating Solutions
 BlackHole Thermodynamics

Cosmology
 The RobertsonWalker Metric
 The Friedmann Equations
 Cosmological Redshift
 Inflation
PHYSICS 366: ADVANCED SOLID STATE PHYSICS
Topics will be selected from the following list:
 Phasetransitions, broken symmetry, collective modes, scaling and renormalizationgroup analysis
 Magnetism:meanfield theory and beyond, itinerant and localized viewpoints, spin waves,ferromagnets and antiferromagnets, spin density waves
 Superconductivity:BCS theory and implications on thermodynamics and transport, gauge invariance,LandauGinzburg theory, electrodynamics
 Disorder: Anderson localization, metalinsulator transitions, interaction effects,Kondo effect
 QuantumHall Effect and correlated electronic systems
 Superfluidityand BoseEinstein condensation
 Physicsof lowdimensional systems: 1Dand 2D systems, surface physics
 Fermiliquid theory
 Quasicrystals
PHYSICS 367: SOFT CONDENSED MATTER PHYSICS
 Granular and Colloidal Matter
 Jamming
 Fluids
 Instabilities
 Topological Shapes and Transitions Between Them
Physics 385: Advanced Math Methods
Main Text: Georgi, Lie Algebras in Particle Physics
Supplementary Texts:
Cornwell, Group Theory in Physics: An Introduction
Tung, Group Theory in Physics
Sternberg, Group Theory and Physics
Gilmore, Lie Groups, Lie Algebras, and Some of TheirApplications

Basic Concepts of Group Theory
 Finite groups
 Representations and reducibility

Lie Groups
 Manifold structure, integration
 Lie algebras
 Global properties; relationship between groups and algebras
 Fundamental and adjoint representations
 SU(2) and its representations

Structure of Lie Algebras
 Roots and weights
 Dynkin diagrams
 Classical groups: SO(n), SU(n), Sp(n), Exceptional groups

Representations
 Tensor methods
 ClebschGordan decomposition
 Young tableaux

Noncompact Groups
 Real and complex forms
 Lorentz group: global structure, discrete subgroups, representations, fermions
 Other noncompact groups
Physics 386: Advanced Methods of Data Analysis
Suggested Texts: TBA
 Probability Distributions
 Covariance and the Propagation of Errors
 Bayesian vs. Frequentist Approaches
 Estimation of Errors
 Curve Fitting and Parameter Estimation
 Fitting in the Presence of Background
 Fitting for a Variance
 Maximum Likelihood Estimators
 Fisher Information Matrix
 Goodness of Fit
 Confidence Intervals
 Time Stream Analysis
 1/f Noise
 Fourier Methods
 MonteCarlo Markov Chains
Physics 443: Quantum Field Theory I
Main Text: M. Peskin and D. Schroeder, Introduction to Quantum FieldTheory
Supplementary Texts:
P. Ramond, Field Theory  a Modern Primer
C. Itzykson and J. Zuber, Quantum Field Theory
S. Weinberg, The Quantum Theory of Fields

Basic field theory
 Representations of the Poincare group
 Dirac equation
 Noether's theorem
 Canonical quantization
 propagators and causal structure
 Interaction picture
 Time ordered products and Wick's theorem

Scattering and Feynman Rules
 LSZ formalism
 Feynman rules for scalar field theory and QED
 Calculation of tree level processes in Φ^{4}and QED
 CPT and spinstatistics

One loop effects
 one loop effects in scalar field theory
 Calculation of in QED
 Unitarity and analytic structure of amplitudes
Physics 444: Quantum Field Theory II
Main Text: M. Peskin and D. Schroeder, Introduction to Quantum FieldTheory
Supplementary Texts:
P. Ramond, Field Theory  a Modern Primer
C. Itzykson and J. Zuber, Quantum Field Theory
S. Weinberg, The Quantum Theory of Fields

Path integral formulation of QFT
 Path integrals for boson fields
 Grassman variables and fermion path integrals
 path integral derivation of Feynman rules
 Ward identities in QED

Renormalization
 Superficial degree of divergence
 explicit oneloop renormalization of scalar field theory
 Wilson's approach to renormalization
 Fixed points and RG flow
 CallanSymanzik equation
 Calculation of beta functions and anaomlous dimensions in simple theories
 Critical phenomena

NonAbelian gauge theory
 Construction of gauge invariant actions
 Feynman rules for gauge theories and FadeevPopov ghosts
 Calculation of asymptotic freedom in QCD
Physics 445: Quantum Field Theory III
Main Text: M. Peskin and D. Schroeder, Introduction to Quantum FieldTheory
Supplementary Texts:
P. Ramond, Field Theory  a Modern Primer
C. Itzykson and J. Zuber, Quantum Field Theory
S. Weinberg, The Quantum Theory of Fields
 Effective potentials and symmetry breaking
 Higgs bosons
 Formulation of electroweak gauge theory
 Coupling to quarks and leptons
 Charged and neutral current processes
 Properties of W and Z bosons
 CKM matrix and CP violation
 Deep inelastic scattering
 Perturbative QCD
 Structure functions and AltarelliParisi equation
Comments:
In most years this course should cover the Standard Model as outlined above. In some years it may cover other advanced topics in QFT such as solitons and instantons, anomalies, large N techniques and lattice gauge theory.