Graduate Course Outlines

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 Physics Department expects that these topics will 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 (Autumn-Winter)
  • 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 (Autumn-Spring)

PHYS 316: Advanced Classical Mechanics

Level: “Classical Mechanics” by Landau & Lifshitz or “Classical Mechanics” by Goldstein, Poole, and Safko.

  1. Variational Principles, Lagrangians, and Hamiltonians
    • Hamilton's principle and Lagrangians
    • Hamiltonians
    • Other variational principles
    • Symmetries and conservation laws
  2. Some Applications
    • Central force problems
    • Rigid body motion
    • Small oscillations
  3. Continuous Systems and Fields
    • Continuum mechanics as a limit of particle systems
    • Lagrangian and Hamiltonian formulations
  4. General Hamiltonian Formulation
    • Poisson brackets
    • Canonical transformations
    • Hamilton-Jacobi theory
    • Action-angle variables
    • Adiabatic invariants
    • Chaos in Hamiltonian systems

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PHYS 322: Advanced Electrodynamics and Optics I

Level: "Classical Electrodynamics" by Jackson

  1. Electrostatics
    1. Gauss' law, potentials
    2. Poisson equation, boundary conditions, methods for solving boundary value problems
    3. Eigenfunction & multipole expansions
    4. Conformal mappings of potentials (optional)
  2. Maxwell's Equations
    1. Potentials, Gauge Transformations
    2. Conservation laws, Poynting vector, Stress-energy tensor
    3. Wave equation
  3. Wave Propagation
    1. Polarization
    2. Dispersive materials, phase/group velocities
    3. Propagation in plasmas, conductors, dielectrics
    4. Waves at media interfaces, Fresnel equations
    5. Waveguides (optional)
    6. Special materials, negative index (optional)
  4. Optics and Diffraction
    1. Diffraction: Fraunhofer and Fresnel
    2. Scattering of plane waves by a conducting sphere
    3. Fourier optics, filtering (see "Intro to Fourier Optics" by Goodman)
    4. Gaussian beams, transfer matrix (optional but recommended)

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PHYS 323: Advanced Electrodynamics and Optics II

Level: "Classical Theory of Fields" by Landau and Lifschitz

  1. Review of Special Relativity; covariant forms
    1. Lorentz covariance of Maxwell's equations
    2. Relativistic action, EM Lagrangian, Hamiltonian
    3. Charged particle action, Lorentz force
  2. Coupling of continuous matter to fields
    1. Dielectrics, thermodynamic properties
    2. Hall effect and thermoelectric phenomena (optional)
    3. Landau-Ginzburg superconductivity (optional)
  3. Radiation
    1. Covariant Green's function, retarded time
    2. Retarded potentials
    3. Multipole radiation
    4. Point currents, scattering plane waves from point charges
    5. Bremsstrahlung, synchrotron, Cherenkov radiation
  4. Special Topics (at instructor's discretion)
    1. Vortices, topological defects in fields/matter
    2. Meta-materials
    3. Physics of charged particle beams, optics of beams
    4. Topological defects
    5. Symmetries of the EM field tensor under dilatation, Noether currents
    6. Electromagnetism and differential geometry

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Physics 330: Mathematical Methods of Physics

Level: Matthews and Walker, Mathematical Methods of Physics or Arfken and Weber, Mathematical Methods for Physicists

  1. Linear Algebra
    1. Finite-dimensional vector spaces
    2. Functional analysis
  2. Complex Analysis
    1. Analytic functions
    2. Contour integration
  3. Ordinary Differential Equations
    1. Exact solutions
    2. Series solutions
    3. Approximation methods
  4. Partial Differential Equations
    1. Hyperbolic, elliptic and parabolic equations
    2. 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.

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Physics 341-342: Quantum Mechanics I-II

Level:
R. Shankar, Principles of Quantum Mechanics
J. Sakurai, Introduction to Quantum Mechanics

  1. Fundamentals of Hilbert Space
    1. Vector spaces and Hilbert spaces
    2. Dirac notation
    3. Self-adjoint and Unitary operators and their spectra
    4. Symmetries and unitary transformations
    5. Projection operators
  2. Simple Quantum Systems and the Relation to Classical Mechanics
    1. Structure of QM
    2. Uncertainty Relations
    3. Two state systems
    4. One-dimensional problems
    5. Coherent States
    6. The classical limit of QM
  3. Time Evolution
    1. Time evolution operator
    2. Heisenberg equations of motion
    3. Heisnberg vs. Schrodinger representation
  4. Symmetry in Quantum Mechanics
    1. Angular Momentum and commutation relations
    2. Spin
    3. Addition of angular momentum
    4. Wigner-Eckart theorem
    5. Identical particle and spin-statistics
  5. Interaction with electromagentic fields
    1. Gauge invariance
    2. Aharonov-Bohm effect
    3. Magnetic monopoles
    4. Stark effect
    5. Landau levels
    6. Quantum Hall effect
  6. Perturbation Methods
    1. Stationary perturbation theory and applications
    2. Time dependent perturbation theory
    3. Fermi's golden rule
    4. Emission and absorption of radiation
  7. Scattering Theory
    1. General formulation for scattering
    2. Cross sections and the scattering amplitude
    3. Definition of S-matrix and analytic properties
    4. Scattering of identical particles
  8. Path Integral Methods
    1. Defining sums over paths
    2. Relation to standard formalism
    3. Phase space path integrals
    4. Evaulating gaussian integrals
    5. Tunneling via instantons
  9. Adiabatic approximation and Berry's phase
    1. Born-Oppenheimer or adiabatic approximation
    2. Sudden approximation
    3. Berry's phase and potential
    4. Global interpretation, examples
  10. Measurement theory and decoherence
    1. Bell's inequalities
    2. Schrodinger's cat and the problem of collapse of the wave function
    3. Decoherence, basic idea and simple models
  11. Supersymmetric quantum mechanics
    1. Supersymmetry
    2. Supersymmetry in quantum mechanics
    3. Solvable examples
    4. 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).

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PHYSICS 352: STATISTICAL MECHANICS

Level: "Statistical Physics of Particles" by Kardar

  1. Thermodynamics
    1. The first and second laws
    2. Carnot engines
    3. Entropy
    4. Approach to equilibrium and thermodynamic potentials
    5. Maxwell relations and stability conditions
    6. The third law
  2. Probability
    1. One random variable
    2. Some important probability distributions
    3. Many random variables
    4. Sums of random variables and the central limit theorem
    5. Rules for large numbers
    6. Information, entropy, and estimation
  3. Kinetic Theory of Gases
    1. Liouville's theorem
    2. The Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy
    3. The Boltzmann equation
    4. The H-theorem and irreversibility
    5. Equilibrium properties
    6. Conservation laws
    7. Hydrodynamic limit
  4. Classical Statistical Mechanics
    1. The microcanonical ensemble
    2. The ideal gas
    3. The canonical ensemble
    4. The Gibbs canonical ensemble
    5. The grand canonical ensemble
    6. Examples
  5. Interacting Particles (optional)
    1. The cumulant expansion
    2. The cluster expansion
    3. The second virial coefficient and van der Waals equation
    4. Mean-field theory of condensation
    5. Variational methods
    6. Corresponding states
    7. Critical behavior
  6. Quantum Statistical Mechanics
    1. Dilute polyatomic gases
    2. Vibrations of a solid
    3. Black-body radiation
    4. Quantum microstates
    5. Quantum macrostates
  7. Ideal Quantum Gases
    1. Hilbert space of identical particles
    2. Canonical formulation
    3. Grand canonical formulation
    4. Non-relativistic gas
    5. The degenerate Fermi gas
    6. The degenerate Bose gas
    7. Superfluid He-4

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PHYSICS 353: ADVANCED STATISTICAL MECHANICS

Level:
"Statistical Physics of Fields" by Kardar
"Scaling and Renormalization in Statistical Physics" by Cardy

  1. Collective Behavior, From Particles to Fields
    1. Phonons and elasticity
    2. Phase transitions in simple systems
    3. Critical behavior
  2. Mean Field Theory
    1. Coarse graining and the Landau-Ginzburg Hamiltonian
    2. Saddle point approximation, mean field theory
    3. Continuous symmetry breaking and Goldstone modes
    4. Discrete symmetry breaking and domain walls
  3. Fluctuations
    1. Scattering and fluctuations
    2. Correlation functions and susceptibilities
    3. Lower critical dimension
    4. Fluctuation corrections to the saddle point
    5. The Ginzburg criterion
  4. The Scaling Hypothesis
    1. Divergence of the correlation length
    2. Critical correlation functions and self-similarity
    3. The renormalization group idea
    4. Block spins
    5. One-dimensional Ising model
    6. Scaling behavior of the free energy
    7. Critical exponents
    8. Scaling for the correlation functions
    9. Scaling operators and scaling dimensions
    10. The Gaussian model
  5. Perturbative Renormalization Group
    1. Diagrammatic representation of perturbation theory
    2. Susceptibility
    3. Perturbative renormalization group
    4. The epsilon-expansion
    5. The Gaussian fixed point, the Wilson-Fisher fixed point
    6. Logarithmic corrections in d = 4
    7. The O(n) model near four dimensions
  6. Lattice Systems (optional)
    1. Models and methods
    2. Transfer matricies
    3. Position space renormalization group in one dimension
    4. The Migdal-Kadanoff bond moving approximation
  7. Series Expansions
    1. Low-temperature expansions
    2. High-temperature expansions
    3. Exact solutions of the 1-dimensional Ising model
    4. Self-duality in the 2-dimensional Ising model
    5. Exact free energy of the square lattice Ising model
    6. Critical behavior of the 2-dimensional Ising model
  8. Low-dimensional Systems and Topological Defects
    1. The nonlinear sigma model
    2. Topological defects in the XY model
    3. Renormalization group for the Coulomb gas
    4. Two-dimensional solids
    5. Two-dimensional melting
  9. Dissipative and Critical Dynamics (optional)
    1. Brownian motion of a particle
    2. Equilibrium dynamics of a field
    3. Dynamics of a conserved field
    4. Non-equilibrium dynamics of open systems
    5. Dynamics of a growing surface
  10. Random Systems (optional)
    1. Quenched and annealed disorder
    2. The Harris criterion
    3. Perturbative approach to the random fixed point
    4. Percolation
    5. Random fields
  11. Conformal Symmetry (optional)
    1. Conformal transformations
    2. Simple consequences of conformal symmetry
    3. The stress tensor
    4. The c-theorem

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PHYSICS 361: SOLID STATE PHYSICS

Level: Ashcroft and Mermin, Solid State Physics

  1. Properties of Insulators
    • Crystal Lattice Structures
    • X-Ray Scattering and Reciprocal Lattice
    • Ground State Properties
    • Lattice Vibrations, Harmonic Theory and Phonons
      • Thermodynamics (Debye Theory)  Spatial
      • Structure (Debye-Waller factor) 
  2. Electronic Properties of Solids
    1. Electrons in a Fixed Periodic Potential (Band Theory)
      • Bloch’s Theorem and Perturbation Theory
      • Tight Binding Systems
      • Density of States
    2. Thermal Properties
      • Insulators and Semi-Conductors
      • Metal
    3. Optical Properties of Solids
    4. Transport in Metals (conductivity, Hall effect, etc.)

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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

  1. Overview
    • Observed particles
    • Forces
  2. Special Relativity and Classical Field Theory Review
    • Spacetime and 4-vectors
    • Relativistic kinematics
    • Field theory: Lagrangians, electromagnetism, gauge invariance
  3. Feynman Diagrams
    • Time-dependent perturbation theory
    • Feynman rules (at tree level)
    • Cross-sections and decay rates
  4. Symmetries
    • Group theory review
    • SU(2) isospin, product representations, SU(3)
    • C, P, and T
  5. Quantum Electrodynamics
    • Spinors and fermions
    • Feynman rules
    • QED processes, Dirac matrix technology
  6. Hadrons and Partons
    • Electron-Quark interactions
    • Inelastic scattering, partons
    • Structure functions
  7. Quantum Chromodynamics
    • Yang-Mills theory
    • Quarks; Feynman rules
    • Running couplings
  8. Electroweak Theory
    • Spontaneous Symmetry Breaking
    • Bosonic sector: vector bosons, Higgs
    • Fermions, SU(2)
    • Generations, CKM matrix, CP violation

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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

  1. Special Relativity
    • Lorentz Transformations
    • Spacetime Diagrams
    • Vectors and Tensors
    • Proper Time
    • Physics in Flat Spacetime
  2. Manifolds
    • Coordinate Systems
    • Vectors as Derivatives
    • Tensor Transformation Law
    • The Metric
    • Tensor Densities
  3. Curvature
    • Covariant Derivatives and Connection Coefficients
    • Parallel Transport
    • Geodesics
    • The Riemann Tensor
    • Geodesic Deviation
  4. Gravitation
    • The Principle of Equivalence
    • Physics in Curved Spacetime
    • Einstein's Equations
    • The Newtonian Limit
  5. Weak Fields and Gravitational Radiation
    • The Weak-Field Limit
    • Linearized Einstein Equations
    • Gravitational Waves
  6. The Schwarzschild Solution and Black Holes
    • Birkhoff's Theorem
    • Geodesics of Schwarzschild
    • Kruskal Extension
    • Penrose Diagrams
    • Charged and Rotating Solutions
    • Black-Hole Thermodynamics
  7. Cosmology
    • The Robertson-Walker Metric
    • The Friedmann Equations
    • Cosmological Redshift
    • Inflation

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PHYSICS 366: ADVANCED SOLID STATE PHYSICS

Topics will be selected from the following list:

  1. Phasetransitions, broken symmetry, collective modes, scaling and renormalizationgroup analysis
  2. Magnetism:meanfield theory and beyond, itinerant and localized viewpoints, spin waves,ferromagnets and antiferromagnets, spin density waves
  3. Superconductivity:BCS theory and implications on thermodynamics and transport, gauge invariance,Landau-Ginzburg theory, electrodynamics
  4. Disorder: Anderson localization, metal-insulator transitions, interaction effects,Kondo effect
  5. QuantumHall Effect and correlated electronic systems
  6. Superfluidityand Bose-Einstein condensation
  7. Physicsof low-dimensional systems: 1Dand 2D systems, surface physics
  8. Fermiliquid theory
  9. Quasi-crystals  

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PHYSICS 367: SOFT CONDENSED MATTER PHYSICS

  1. Granular and Colloidal Matter
  2. Jamming
  3. Fluids
  4. Instabilities
  5. Topological Shapes and Transitions Between Them

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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

  1. Basic Concepts of Group Theory
    • Finite groups
    • Representations and reducibility
  2. Lie Groups
    • Manifold structure, integration
    • Lie algebras
    • Global properties; relationship between groups and algebras
    • Fundamental and adjoint representations
    • SU(2) and its representations
  3. Structure of Lie Algebras
    • Roots and weights
    • Dynkin diagrams
    • Classical groups: SO(n), SU(n), Sp(n), Exceptional groups
  4. Representations
    • Tensor methods
    • Clebsch-Gordan decomposition
    • Young tableaux
  5. Noncompact Groups
    • Real and complex forms
    • Lorentz group: global structure, discrete subgroups, representations, fermions
    • Other noncompact groups

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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
  • Monte-Carlo Markov Chains

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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

  1. 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
  2. Scattering and Feynman Rules
    • LSZ formalism
    • Feynman rules for scalar field theory and QED
    • Calculation of tree level processes in Φ4and QED
    • CPT and spin-statistics
  3. One loop effects
    • one loop effects in scalar field theory
    • Calculation of in QED
    • Unitarity and analytic structure of amplitudes

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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

  1. 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
  2. Renormalization
    • Superficial degree of divergence
    • explicit one-loop renormalization of scalar field theory
    • Wilson's approach to renormalization
    • Fixed points and RG flow
    • Callan-Symanzik equation
    • Calculation of beta functions and anaomlous dimensions in simple theories
    • Critical phenomena
  3. Non-Abelian gauge theory
    • Construction of gauge invariant actions
    • Feynman rules for gauge theories and Fadeev-Popov ghosts
    • Calculation of asymptotic freedom in QCD

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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

  1. Effective potentials and symmetry breaking
  2. Higgs bosons
  3. Formulation of electroweak gauge theory
  4. Coupling to quarks and leptons
  5. Charged and neutral current processes
  6. Properties of W and Z bosons
  7. CKM matrix and CP violation
  8. Deep inelastic scattering
  9. Perturbative QCD
  10. Structure functions and Altarelli-Parisi 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.

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