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Syracuse University, College of Arts and Sciences

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, B fields Coulombs law and multi-pole expansions Magnetic dipole moment
  • 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 Jz
    • Spherical harmonics Rotationally invariant problems
      • Free particle in 3D
      • Particle in spherical "square" well
      • Isotropic harmonic oscillator
  • Spin
    • Pauli matrices; Eigenstates of Sň.
    • Particle with magnetic moment in magnetic field
    • Magnetic resonance
  • 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, 2nd 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)
  • 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*