The graduate curriculum is based upon a "core" of six courses for which all incoming graduate students must pass final exams at the 60% level. The core courses taken in the first semester are: Mathematical Methods in Physics; Dynamical Systems (Classical Mechanics); and Quantum Mechanics I. In the second semester, graduate students take: Quantum Mechanics II; Thermal and Statistical Physics; and Advanced Classical Electricity and Magnetism. The final exams in these courses constitute the Comprehensive Examination in Physics, which must be passed in order for a student to proceed to Ph.D. candidacy. This document is designed to facilitate a uniform standard for core courses from year to year. Textbook recommendations are made only in an effort to aid in establishing the level of the courses. No specific textbook is required.

As it is the final examinations in these courses that serve as the Comprehensive Examination and for which uniformity is the highest priority, the standardization is specified as follows. For each core course, at least 80% of the points available on the final examination must be based upon the topics listed in the following standards. Examinations should be designed such that a grade of 60% is the level expected of a successful Ph.D. student and such that students that have mastered the material in a standard textbook (such as one of the suggested textbooks) on the subject can successfully pass the examination.

## Mathematical Methods

• Linear Algebra: Vector Spaces; Basis Transformations; Matrix Diagonalization.
• Calculus of Multiple Variables: Differential Vector Operators; Line, Surface, and Volume Integrals; Conservative Fields
• Ordinary Differential Equations: Basic Methods; Series Solutions; Green's Functions
• Transforms: Fourier Series; Fourier Transforms; Laplace Transforms
• Complex Analysis: Analytic Functions; the Cauchy and Residue Theorems; Laurent Series
• Partial Differential Equations: Separation of Variables; Green's Functions; Green's Functions for Dirichlet & Neumann Problems
• Calculus of Variations: Variations with Constraints; Eigenvalue Problems
• Linear Differential Operators: Self-adjoint Operators and Boundary Conditions, Eigensystems, and Hilbert Spaces

## Dynamical Systems

• Lagrangian and Hamiltonian Mechanics Including Canonical Variables, Systems Subject to Constraints, Integrals of Motion, and Poisson Brackets
• Oscillations, Normal Modes, and Periodically-driven Systems
• Rotating Systems and Rigid Body Motion
• Waves
• Relativistic Mechanics
• If used, textbooks at the level of the following (designated by author) or similar are recommended: Goldstein, Poole, & Sarko; Fetter & Walecka.

## Quantum Mechanics I

• Formal Framework Including: Hilbert Space, Operators and States; Probabilities; Commutation Rules; The Uncertainty Principle; The Schrodinger & Heisenberg Pictures
• Basic Systems: Motion in a Central Potential; Motion in a Magnetic Field; The Harmonic Oscillator
• Continuous & Discrete Symmetries: Translations; Rotations; Gauge Invariance; Parity; Time Reversal; Symmetry Groups (including specifically the rotation group); Symmetry and Conservation Laws
• Multiple Particle States: Superposition; Entanglement; Mixing; The Density Matrix
• The Theory of Angular Momentum: The Rotation Group; Spin; Addition of Angular Momentum; Representations of the Rotation Group; Vector & Tensor Operators; The Wigner-Eckart Theorem
• If used, textbooks at the level of the following (designated by author) or similar are recommended: Sakurai; Merzbacher; Gottfried & Yan; Schiff; Messiah

## Quantum Mechanics II

• Approximation Methods: Perturbation Theory (both time independent and time dependent); Standard Applications Including Atomic Fine Structure, the Zeeman Effect, and the Stark Effect.
• Scattering: Partial Wave Analysis: Green's Functions; the Analytical Structure of Scattering Amplitudes; the Born Approximation; the Optical Theorem; the Scattering Matrix
• Path Integral Formulations and the Aharonov-Bohm Effect.
• Quantization of the Electromagnetic Field; Creation and Annihilation Operators; Radiative Transitions; Fermi's Golden Rule.
• Introduction to Relativistic Quantum Mechanics.
• If used, textbooks at the level of the following (designated by author) or similar are recommended: Sakurai; Merzbacher; Gottfried & Yan; Schiff; Messiah; Bohm.

## Advanced Classical Electricity & Magnetism

• Basic Electrostatics: Poisson & Laplace Equations; Methods of Solution for Poisson/Laplace Equations; Green's Functions; Method of Images; Multipole Expansions; Boundary Conditions at Material Transitions.
• Basic Magnetostatics: The Bio-Savart & Ampere Laws; the Vector Potential; Multipole Expansions; Localized Current Distributions; Boundary Conditions at Material Transitions.
• Electrodynamics Including: Potentials & Gauge Transformations; The Poynting Theorem & Momentum Conservation; Plane & Spherical Waves, Polarization, Reflection, Transmission, and Refraction; Propagation of Light in Media; Dispersion Relations; Basics of Waveguides.
• Radiation: Radiation from a Localized Source; Radiation from a Moving Point Source and the Lienard-Wiechert Potentials; Scattering.
• Special Relativity: The Minkowski Four-Vector Formalism; Lorentz Transformations; Invariance and Covariance; Scalars, Vectors, and Tensors; Relativistic Mechanics; the Maxwell Equations in Covariant Form; the Stress-Energy Tensor; Lagrangian & Hamiltonian Formulation of Relativistic Charged Particles and the Electromagnetic Field.
• If used, textbooks at the level of the following (designated by author) or similar are recommended: Jackson; Panofsky & Philips.

## Thermal and Statistical Physics

• Thermodynamics: Energy; Entropy; the Four Laws of Thermodynamics; Thermodynamic Potentials; Response Functions, Especially Heat Capacities; Maxwell Relations; Equilibria of Systems with Various Constraints; Heat Engines; Phase Transitions and the Clausius-Clapeyron Relation.
• The Microcanonical Ensemble: Phase Space & Liouville's Theorem: Maximization of Entroy and Equilibria in the Microcanonical Ensemble; Equipartition; the Ideal Gas & the Gibbs Correction.
• The Canonical Ensemble: Equilibria in the Canonical Ensemble; the Partition Function; the Ideal Gas; Paramagnetism; Two-state Systems.
• The Grand Canonical Ensemble: Equilibria in the Grand Canonical Ensemble; the Grand Partition Function; Chemical Potentials.
• Quantum Statistics: The Density Matrix; the Connection Between Spin and Statistics.
• Ideal Bose Systems: Blackbody Radiation; Sound Waves; the Debye Model; Bose-Einstein Condensation.
• Ideal Fermi Systems: Electrons in Metals; Diamagnetism; Degeneracy.
• If used, textbooks at the level of the following (designated by author) or similar are recommended: Pathria & Beale; Huang; Callen; Kardar.