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 I (Autumn)
  • Physics 341, 342 - Quantum Mechanics I and II (Autumn-Winter)
  • Physics 352 - Statistical Mechanics (Spring)
  • 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)
  • Physics 471 – Introduction to Modern Atomic Physics (Winter/Spring)

PHYSICS 316: ADVANCED CLASSICAL MECHANICS

Level: “MathematicalMethods of Classical Mechanics” by Arnold or “Classical Mechanics” by Landau & Lifshitz.

A. Variational Mechanics

  1. Calculus of variations and variational principles 
  2. Hamilton equations of motion
  3. Principle of least action
  4. Canonical transformations; Poisson brackets and the Jacobi identities
  5. Poisson brackets and commutators
  6. Hamilton-Jacobi theory. Principal function and characteristic function
  7. Action-angle variables
  8. Small oscillations

B.Continuous Systems with Infinite Degrees of Freedom

  1. Lagrange and Hamiltonian formulations
  2. Hamiltonian canonical equations
  3. Density conservation laws
  4. Integral conservation laws and Poisson brackets
  5. Transition to quantum mechanics

C. Symmetries and Conservation Laws

  1. Noether's theorem and applications
  2. Scale invariance
  3. Accidental degeneracies and their corresponding symmetries
  4. Adiabatic Invariants

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PHYSICS 322: ADVANCED ELECTRODYNAMICS AND OPTICS I

Level: Classical Electrodynamicsby Jackson

Electromagnetic Fields and RelativisticParticles by Konopinski
The Classical Theory of Fieldsby Landau and Lifschitz

Prerequisite: Physics 330

  1. Maxwell's Equations
    1. Definitions of the vectors E,D, P, and B, M, H.
    2. Units
    3. Integral and differential forms of Maxwell's Equations
    4. Representation of E and B in terms of vector Potential A and Scalar Potential Φ.
    5. Gauge Transformations, Lorentz gauge, Coulomb gauge.
    6. Wave Equations for E, B, A and θ. Retarded integrals
    7. Energy, momentum and stress Tensors. Poynting vector, etc.
    8. Magnetohydrodynamic limit.
  2. Plane Electrodynamic Waves
    1. Polarization. Form of E andB. Forms of θ, A.
    2. Hertz vector in vacuum, dielectric, magnetic, dispersive media.
    3. Phase and group velocities
    4. Conducting medium, dissipation, ground wave.
    5. Reflection, refraction, boundaries.
    6. MHD waves, cold plasmas, Faradayrotation.
  3. Transmission Lines, Wave Guides,Resonant Cavities
    1. Cylindrical and rectangularboundaries
    2. Energy flow, losses.
    3. Dielectric waveguides, fiber modes
  4. Radiation
    1. Expansion of retarded integralsfor non-relativistic case
    2. Dipole, quadrupole radiation.Near and far fields.
    3. Cyclotron radiation, center-fedlinear antenna
    4. Thomson scattering. Scatteringby conducting needles
    5. Small dielectric spheres.
  5. Diffraction.
    1. Discussion of rigorous solution
    2. Huygen's principle and Kirchoff'sintegral
    3. Fraunhofer and Fresnel diffraction
    4. Rectangular and circular apertures.
    5. Fresnel lens, Babinet's principle.

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PHYSICS 323: ADVANCED ELECTRODYNAMICS AND OPTICS II

Level: Classical Electrodynamics by Jackson

Electromagnetic Fields and Relativistic Particles by Konopinski
The Classical Theory of Fields by Landau and Lifschitz

Prerequisite: Physics 322

  1. Review of Special Relativity
    1. Lorentz transformations of coordinates,field components
    2. Four vectors and tensors. Covarianceof Maxwell's equations.
    3. Transformation of moving electrically-neutral current-carrying conductor
    4. Covariant Lorentz force.
  2. Motion of Charged Particlesin Electromagnetic Fields.
    1. Equations of motion, propertime, covariant forms.
    2. Motion in crossed electric andmagnetic fields
    3. Motion in time-varying magneticfield, in plane wave
    4. Invariants, of particle motion, guiding center approximation
    5. Collisionless plasma equationsas sum over individual particle motions
    6. Chew-Goldberger-Low approximation
  3. Collisions Between ChargedParticles
    1. Scattering, energy loss, Coulombcollisions.
    2. Equipartition or equilibrationtime for individual particles undergoing Coulomb collisions with background. Mean free path
    3. Scattering of fast particles byatoms
    4. Multiple scattering
  4. Radiation by Moving and Interacting Charges
    1. Dipole radiation
    2. Lenard-Wiechert potentials
    3. Synchrotron radiation
    4. Brehmstrahlung
    5. Thomson scattering
    6. Cerenkov radiation
    7. Radiation reaction, radiativedamping
    8. Scattering and absorption
    9. Line width and level shift ofoscillators
    10. Self-energy and self-momentumof charged particle

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Physics 330: Mathematical Methods of Physics I

Main Text: Matthews and Walker, Mathematical Methods of Physics
Supplementary Text: Arfken and Weber, Mathematical Methods for Physicists

  1. Complex Analysis
    • Analytic functions
    • Contour integration
  2. Ordinary Differential Equations
    • Exact solutions, special functions
    • Series solutions
    • Approximation methods (WKB, perturbation theory)
  3. Linear Algebra
    • Vector spaces and matrices
    • Infinite-dimensional spaces; Fourier and other transforms
  4. Partial Differential Equations and Boundary Value Problems
    • General properties
    • Green's functions
    • Boundary-value problems

Note: This outline is intended as a guide to the most essentialtopics for this course; there is some flexibility in the order and mannerof presentation. In particular, examples of particular applications areleft to the instructor.

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Physics 385: Mathematical Methods of Physics II or 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 341-342: Quantum Mechanics I-II

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

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

Comments:

The first seven items should be considered the core material, most of which should be covered in the course. There should usually be time tocover some of the remaining four topics (or other topics chosen by the instructor).

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PHYSICS 352: STATISTICAL MECHANICS

Level: Equilibrium Statistical Mechanics by Mazenko, or Statistical Mechanics by Pathria

I. General Principles of Statistical Mechanics

  1. Maximum Entropy, the Second Law of thermodynamics, and Equilibrium
  2. Microcanonical Ensemble
  3. Open Ensembles
    • CanonicalEnsemble and temperature
    • GrandCanonical Ensemble and chemical potential
  4. Fluctuations
  5. Symmetry and equilibrium ensembles
  6. Mechanical forces: solids and liquids

II. Principles of Thermodynamics

  1. General postulates of thermodynamics
  2. Thermodynamic transformation theory
    • Legendretransforms
    • Maxwellrelations
    • Jacobians
  3. Fluctuations and stability
    • Phaseequilibrium
    • Mixtures
    • Chemicalreactions
  4. Landau theory of phase transitions
    • Orderparameters
    • Effectivefree energy
    • Criticalpoints
  5. van der Waal's equation of state

III. Quantum Statistical Mechanics

  1. Statistical mechanics in the language of Second Quantization
  2. Ideal quantum systems
    • Densityof states
    • Classicallimit
    • Black-bodyradiation, the photon gas
    • IdealFermi systems at low temperatures
  3. Bose-Einstein condensation

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PHYSICS 361: SOLID STATE PHYSICS

Level: Ashcroft and Mermin, Solid State Physics

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

Level: “Principles of Condensed Matter Physics” by Chaikin and Lubensky

  • Structure and scattering
  • Thermodynamics and statistical mechanics
  • Mean-field theory
  • Field theories, critical phenomena, renormalization group approaches
  • Generalized elasticity
  • Dynamics: correlations and response
  • Hydrodynamics
  • Topological defects
  • Walls, kinks, solitons
  • Atthe discretion of the instructor, additional topics covered may include colloids, liquid crystals, complex physics, polymer physics.

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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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Physics 471: Introduction to Modern Atomic Physics

In this course, a selection of current research topics in the field of atomic physics will be explored. Previous exposure to an atomic physics course is preferred, but not required.

There will be no designated text books. However, you may find the following references helpful:

  1. Physics of Atoms and Molecules, by B. H. Bransden and C. J. Joachain;
  2. Atomic Physics, by D. Budker, D. F. Kimball and D. P. DeMille;
  3. Students are expected to read several original papers on each subject.

Outline:

  1. Nuclear Magnetic Resonance
    • Two-level quantum system
    • Bloch vector
    • Rabi oscillation
  2. The Hydrogen Atom
    • Theory: Bohr-Dirac-Schwinger
    • Laser spectroscopy on the 1S-2S transition
    • Positronium, muonium, and anti-hydrogen
  3. Trapping and Cooling
    • Ions: Paul trap and Penning trap
    • Neutral atoms: magneto-optical trap and optical dipole trap
  4. Atomic Clocks
    • Cesium microwave clocks: beam and fountain
    • Optical clocks
  5. The Fine Structure Constant
    • Fine structure of hydrogen and helium atom
    • The anomalous magnetic moment of electron
    • The constancy of the constant
  6. Probing Atomic Nuclei
    • Nuclear moment effects
    • Nuclear size effects
    • Spectroscopy of rare isotopes
  7. Fundamental Symmetries in Atoms
    • Parity violation
    • Time reversal invariance: atomic EDM
    • Lorentz invariance
  8. Quantum Degenerate Atomic Gases
    • Bose-Einstein condensate
    • Fermionic degenerate system
  9. Quantum information
    • Quantum entanglement
    • Encryption and computation

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