# Graduate Syllabus

## Key to annotations:

- No further annotation fully discussed in coursework (lecture, assigned readings, and/or assigned problems).
- The asterisk
^{*}denotes topics discussed in coursework, but may appear on exam in lesser sophistication than other topics.

## Classical Mechanics

Goldstein, Poole, and Safko (Classical Mechanics, third edition) and Strogatz (Nonlinear Dynamics and Chaos).

- Lagrange's Equations and d'Alembert's Principle
- Electromagnetic potentials
- Applications

- Variational Principles
- Calculus of variations
- Nonholonomic systems and constraints
- Conservation Theorem

- Rigid body motion
- Theory of rotations, finite and infinitesimal
- Euler angles
- Coriolis effect
- Inertia Tensor
- Applications

- Small oscillations
- Eigenvalue problem
- Normal modes and frequencies

- Hamilton's Equations and Hamilton Jacobi Equation
- Variables for Hamilton's Equations
- Canonical transformations via generating functions Applications of Hamilton-Jacobi
- Time-dependent perturbation theory

- Nonlinear Equations
- One dimensional equations and Bifurcations
- Linear and nonlinear equations in two dimensions; limit cycles

## Electricity and Magnetism

References:

Jackson (Classical Electrodynamics) and Landau-Lifshitz (The Classical Theory of Fields).

- Introductory Mathematics
- Elements of distribution theory and the δ-function
- Four-vector notation
- Maxwell and Lorentz-force Equations in 4-vector and component forms.
- Integral forms oaf Maxwell's equations Magnetic monopoles
- Gauge transformation and gauge fixing
- Typical vector potentials for prescribed
,*E*.*B* - Energy and its conservation law; Poynting vector
- The action principle

- Simple Problems of Charged Particles in External Fields
- Motion of charged particle in static
,*E*fields Coulombs law and multi-pole expansions Magnetic dipole moment*B*

- Motion of charged particle in static
- Wave Propagation
- Green's Functions for Poisson's Equation
- General theory
- Method of images
- Separation of variables

- Dielectrics: Definitions and Boundary Value Problems
- Magnetostatics
- Concluding lectures
- Magetic monopoles
- GUTS and proton decay
- Parity and time reversal

- Quantum Mechanics
- Linear Vector Space Formalism
- Linear dependence, dimension, basis
- Inner product, operators, matrix isomorphism
- Hermitian and unitary operators
- Diagonalization. infinite dimensional LVS's

- Postulates
- Connection between physical system and LVS
- Position and momentum operators
- Measurement
- Dynamics

- Solvable problems in one dimension
- Free particle Square well
- Scattering and tunneling
- Linear harmonic oscillator
- Solution in coordinate representation
- Solution with raising/lowering operators

- Hydrogen atom radial equation

- Miscellaneous topics
- Ehrenfest's theorem Uncertainty principle
- Generalization to
*N*degrees of freedom - Identical particles
- Addition of angular momentum
- Irreducible tensor operators and the Wigner-Eckart theorem

- Scattering theory
- Born approximation
- Partial wave analysis

- Symmetries
- Translations in space and time
- Reflection of coordinates and parity
- Rotations, generators of rotations, and angular momentum
- Vector operators
- Simultaneous operators of
**J**and*J*_{z} - Spherical harmonics Rotationally invariant problems
- Free particle in 3D
- Particle in spherical "square" well
- Isotropic harmonic oscillator

- Spin
- Pauli matrices; Eigenstates of
•*S**ň*. - Particle with magnetic moment in magnetic field
- Magnetic resonance

- Pauli matrices; Eigenstates of
- Approximations
- Time independent perturbation (including degeneracy)
- WKB approximation
- Variational method
- Time-dependent perturbation theory, application to radiation

- Relativistic quantum mechanics
- Free Klein Gordon field
- Quantization of free photon
- Dirac equation; properties and uses

## Statistical Mechanics and Thermodynamics

References: Unless otherwise noted, the main bibliographic source is: Pathria, Statistical Mechanics, 2^{nd} edition

- Elements of Thermodynamics (refs: Goostein, Reif, Pathria)
- Four Laws of Thermodynamics
- Carnot Engine
- Ideal Gas, van der Waals Gas
- Thermodynamic Potentials and Response Functions
- Manipulation of thermodynamics quantities

- Elements of Probability
^{*}(ref: Reif)- Binomial distribution
- Poisson distribution
- Central Limit Theorem, Normal distribution

- Statistical Mechanics: Ensemble Theory; connection with thermody- namic quantities and applications
- Microcanonical Ensemble
- Canonical Ensemble
- Grand Canonical Ensemble

- Statistical Mechanics: Density Matrix Formulation
^{*} - Statistical Mechanics: Classical Ideal Gas
- Statistical Mechanics: Fermi-Dirac and Bose-Einstein Statistics; Quan tum Gases
- Bose Gas & Bose-Einstein condensation
- Superfluids: mostly LHe phenomenology (refs: Huang, Good stein)
- Blackbody Radiation
- Phonons (Debye model)

- Fermi Gas (applications: spin systems, metals, stars)

- Bose Gas & Bose-Einstein condensation
- Statistical Mechanics: Corrections to the Ideal Gas Law
^{*} - Statistical Mechanics: Phase Transitions (refs: Pathria, Plischke & Bergersen, Goodtein, Huang)
- Thermodynamics
- Mean Field Theory (applications: Ising, van der Waals)
- Landau theory
- Critical exponents and scaling
^{*}