# Qualifier Exam

## Information on the qualifying exams

### Time and Place

The exams will take place on Friday, May 8, 2015 from 1 to 4pm (CM, SM) and on Monday, May 11, from 9am to 12pm (EMI, QMI), and continuing from 1 to 4pm (EMII and QMII or Astro I and II, depending on the track). All exams will take place in Phillips room 265. During the exams you are free to briefly leave the room. There will also be drinks and snacks provided.

### Subject and Format

The material on which you will be tested is from the 1st year courses, and the test problems will adhere to the corresponding topics (see below). Each of the 6 exams will consist of 5 problems, of which you’ll have to pick out and solve 3. Solving any additional problems is ok – but you’ll be credited for the 3 with the highest scores. Partial credits for any attempted solutions will be given, too.

### Allowed Materials

Bring a writing implement. Exam worksheets will be provided that have headers for identification (see below), problem section and number, and page number. You are welcome to bring your own scrap paper. You can also bring integral tables and a calculator. For the astro exams the list of pertinent data and constants will be provided.

### Protocols

There will be a number assigned to each test, and shown on the first page. Sign and write your name and PID on the front page of each exam. Also, write the number assigned to the test (but not your name or PID in order to ensure the anonymity of grading) on each page of your solutions which have to be marked, according to the number of the section, problem, and page (example: EMI, problem 2, page 3). Write legibly.

### Grading

Grading of the exams typically takes about 2 weeks, after which period the results will be discussed at a faculty meeting and the decision on passing/failing will be made. All the students will be individually informed about those decisions by the Graduate Studies Chair shortly thereafter.

### Grading Rubric

10 points | Solution is impeccable, including relevant physical comments |

9 points | Nearly impeccable solution, lacking only some physical discussion or some very minor detail (minor numerical pre-factor error, inconsequential sign error, some but not all relevant discussion of e.g., conservation, symmetries, boundary conditions, etc.) |

7-8 points | Strong partial solution, missing minor elements; lacking correct physical discussion (failed to answer part of a multistep problem, mathematical solution but lacking significant discussion) |

6 points | A sufficient response to pass, with headway toward solution but missing some key elements (passing but barely because of correctly answering part of the problem but failing to respond to another part or giving an incorrect response to part of the problem) |

5 points | On the exact border of being acceptable or not as a passing/failing response |

4 points | Not a passing response but one with some correct headway on a part of the problem |

3 points | Basic formulas and some comments/discussion about how to solve the problem |

1-2 points | Basic formula(s) written down which are at least tenuously related to the problem |

0 points | Blank sheet or one with essentially no relevant content |

### Topic Lists

**Classical mechanics** (Khveshchenko)

- Lagrangian dynamics

Variational principle, action, and Lagrangian. Lagrange-Euler equations, holonomic constraints, and Lagrange multipliers. Noether theorem: symmetries, and integrals of motion. - Hamiltonian dynamics

Hamilton equations, Legendre transformations, cyclic coordinates, and conserved momenta. Canonical transformations and generating functions, Poisson brackets and symplectic structures, Liouville theorem, Hamilton-Jacobi equation and action-angle variables, adiabatic invariants. Canonical perturbation theory: time-dependent and -independent versions. - Applications to mechanical systems

Central force problem: virial theorem, scattering theory, integrals of motion and stable orbits for the Kepler potential. Rigid body rotations: angular momentum, inertia tensor, Euler angles, centrifugal and Coriolis forces, torque-free and heavy tops. Harmonic oscillations: eigen frequencies and normal coordinates, stability, effects of dissipative and driving forces. - Relativistic kinematics

Lorentz transformations, Minkowski space-time. Basic effects of special relativity. 4-vectors (velocity, momentum, force), tensors (electromagnetic field, metric, stress-energy), covariant equations of motion. Relativistic collisions and cross-sections. Classical field theory: Lagrangian and Hamiltonian formulations.

**Statistical mechanics** (Wu)

- Thermodynamics

Equilibrium and State Quantities. The Laws of Thermodynamics. Phase Transitions and Chemical Reactions. Thermodynamic Potentials. - Statistical Mechanics

Number of Microstates Ω and Entropy S. Ensemble Theory and Microcanonical Ensemble. The Canonical Ensemble. Applications of Boltzmann Statistics. The Macrocanonical Ensemble. - Quantum Statistics

Density Operators. The Symmetry Character of Many-Particle Wavefunctions. Grand Canonical Description of Ideal Quantum Systems. The Ideal Bose Gas. The Ideal Fermi Gas. Real Gases. Classification of Phase Transitions. The Models of Ising and Heisenberg.

**Electromagnetism I** (Drut)

- Maxwell’s equations

Elementary properties. Linearity. Regime of validity. Helmholtz theorem. - Statics in vacuum

Electrostatics. Charge distributions. Delta function, Heaviside function. Poisson equation. Laplace equation. Dirichlet & Neumann boundary conditions. Uniqueness theorem. Electrostatic energy. Multipole expansion of charge distributions, potentials and energy. Maxwell stress tensor. - Methods of solution

Problems in 2D & 3D. Method of images. Green’s functions and Green’s identities. Separation of variables. Special functions for cartesian, cylindrical and spherical coordinates. Conformal mapping. - Statics in materials

Dielectrics & conductors. Boundary conditions. Polarization & Displacement vectors. Ferromagnets. Vector potential and magnetic scalar potential in magnetostatics. Linear vs. non-linear media. - Quasistatic fields

Current distributions. Magnetic field due to a localized current distribution. Faraday’s law of induction. Energy in the magnetic field. - Dynamics I

Vector and scalar potentials. Gauge transformations. Wave equation. Green’s functions for the wave equation. Macroscopic electromagnetism. Boundary conditions. Poynting’s theorem. Reflection and refraction at boundaries. - Dynamics II

Frequency dispersion in dielectrics, metals and plasmas. Causality and Kramers-Kroenig relations. Elementary magnetohydrodynamics. Wave guides. Resonant cavities. Radiation by a localized oscillating source. Electric dipole and magnetic quadrupole fields.

**Electromagnetism II** (Evans)

- Time-dependent Maxwell’s equations, Green functions, gauge, Poynting theorem

Potentials and gauge freedom. Green functions and integral solutions. Energy, momentum, stress of EM field. Poynting theorem. - Plane waves, polarization, dispersive media

Plane wave solutions. Polarization, linear modes, circular modes, elliptic modes. Spectral content and partial polarization. Coherence and incoherence. Dispersive media, waves in plasma, reflection and transmission. Causality and green functions, absorption and anomalous dispersion. - Wave guides and resonant cavities

Radiation from isolated source, multipole expansions. Wave generation from isolated, slow-motion sources, multipole expansion. Larmor’s formula for electric dipole radiation. Thomson scattering. Electric quadrupole and magnetic dipole radiation. - Special relativity, Lorentz covariance, relativistic kinematics

Special relativity, covariance, frame transformations, four velocity, acceleration. Lorentz invariants, relativistic kinematics, particle scattering processes. Boosts, Doppler shift, aberration, and beaming. Four potential, four current, Maxwell’s equations in relativistic form, gauge condition. Field strength tensor, dual field strength tensor, Maxwell’s equations in 1st order form. Transformation of E and B. Lorentz force. Emitted flux and received flux. - Radiation from a relativistic point charge

Relativistic point charge, Lienard-Weichart potentials, E and B field of relativistic charge. Radiated flux, integral over past light cone and Fourier decomposition. Synchrotron radiation, cyclotron radiation. Bremsstrahlung radiation. Appearance, disappearance, and transition radiation. Cherenkov radiation

**Quantum mechanics I** (Engel)

- Formalism

Dirac notation for vectors and functionals. Linear algebraic relations in different bases (position, momentum, z-spin, x-spin…). Linear operators, Hermitian operators, unitary operators. Translation operator and relation with momentum. Measurement and uncertainty relations. Gaussian wave packets. - Time Evolution and Energy Eigenstates

Schrodinger picture and time-evolution operator. Connection between Hamiltonian eigenvalue problem and time evolution. Energy-time uncertainty relation. Time evolution in vector spaces with dimension two. Heisenberg picture. Simple Harmonic Oscillator: eigenstates, spectrum, raising and lowering operators, time evolution in both pictures. Coherent oscillator states. Schrodinger equation in position and momentum space. Simple 1-d potentials. Schrodinger equation in electromagnetic field. Aharanov-Bohm effect and magnetic monopoles. Elementary concepts for path integrals. Semiclassical limit and WKB approximation. - Rotation and Angular Momentum

Angular momentum as generator of rotations. Rotation matrices for spin-1/2 systems. Euler-angle parameterization of rotations. Basics of density operators and ensembles. Angular momentum eigenvalues and eigenstates, raising and lowering operators. Orbital angular momentum and Schrodinger equation in 3d. Addition of angular momenta. Bell’s inequality for correlated spins. Spherical-tensor operators. Wigner-Eckart theorem and reduced matrix elements.

**Quantum mechanics II** (Mersini)

- Symmetry in Quantum Mechanics

Symmetries, conservation laws, degeneracies. Discrete symmetries, parity, space inversion. Time-reversal discrete symmetry. - Approximation Methods

Perturbation theory and variational methods - Time-Independent Perturbation Theory

Nondegenerate case. Degenerate case. Hydrogen like atom: fine structure and Zeeman effect. Variational methods. Time dependent potentials: interaction picture vs. Heisenberg picture. Time dependent perturbation theory. Application to interaction with classical fields. Energy shifts and decay width - Identical Particles

Permutation symmetry. Symmetrization postulate. Two electron system - Scattering Theory

Lippmann-Schwinger equation. Born approximation. Optical theorem. Eikonal approximation. Free plane wave vs. spherical wave. Method of partial waves. Low energy scattering and bound states. Resonance scattering. Identical particles and scattering. Symmetry considerations in scattering. Time dependent formulation of scattering. Inelastic scattering. Coulomb scattering.

**Stellar astrophysics** (Clemens/Cecil/Champagne)

- Observational Matters

Luminosity-flux-distance relation, luminosity-temperature-radius relation, magnitudes. The Hertsprung Russell diagram. Clusters and stellar populations in galaxies. Stellar mass and Kepler’s Law. - Interior Structural Equations

Mass conservation. Hydrostatic equilibrium. Stellar dimensional analysis (homology). Timescales (dynamical, thermal, evolutionary). Polytropes - Interior energy and transport equations

Boltzmann and Saha Equations. Equations of State (including degeneracy). Radiative energy transport in the stellar interior (opacities). Convective energy transport - Atmospheres

Frequency dependent radiative transport. Gray atmospheres. Limb darkening. Curves of growth. - Nuclear processes

Stellar reaction networks. Gamow peak. R and S processes. - Stellar structure and evolution models

Static and quasistatic models. Evolutionary sequences and comparison to H-R diagram. Stellar pulsations.

**Cosmology** (Erickcek)

(This section relevant only when the core course in cosmology is taught)

- The Expanding Universe

Metrics, geodesics, curvature tensors, Einstein field equations. The Friedmann-Robertson-Walter metric and Friedmann equations. Conservation laws and evolution of energy density. Cosmological redshift. Distance measures. - Early-Universe Thermodynamics

Density, pressure, and entropy in the early Universe. Big Bang nucleosynthesis. Recombination and the cosmic microwave background. Thermal relics. - Inflation and the Origin of Inhomogeneity

Motivation for inflation. Scalar field dynamics. Slow-roll inflation. Generation of perturbations during inflation. The primordial power spectrum of curvature and tensor perturbations. - Cosmological Perturbation Theory

Metric perturbations, gauges, and linearized Einstein equations. Boltzmann equations. Evolution of cosmological perturbations. The matter power spectrum. Anisotropies in the cosmic microwave background. - Large Scale Structure and Dark Energy

Spherical collapse. Mass variance. Press-Schechter formalism. Halo bias. Halo properties and models. Velocity fields. Redshift space distortions. Observational evidence for dark energy. Dark energy models.

**Astro-II: High Energy** (Evans)

(This section relevant only when the core course in high energy astrophysics is taught)

- High-density and relativistic stellar interiors: application to white dwarfs and neutron stars

Polytropes. Degenerate electron gas, special relativity. Stellar structure and white dwarf stars. Beta decay, photo-disintegration. Nuclear statistic equilibrium. Chandrasekhar mass and collapse. Neutrino trapping. Tolman-Oppenheimer-Volkoff equations and general relativity. Neutron star structure, n/p ratio. - Radiative processes, special relativity, electromagnetic theory: application to pulsars

Special relativistic covariance and electromagnetism. Electric dipole and Larmor’s formula. Magnetic dipole radiation. Pulsar phenomenology, braking, glitches. Synchrotron radiation and inverse Compton scattering. P–P dot diagram. Emission mechanism. - Hydrodynamics, black holes: application to compact accretion sources

Accreting X-ray sources, LMXB, HMXB, AGN. Schwarzschild and Kerr black holes. Relativistic orbits and energy conversion. Accretion disks, hydrodynamics, radiative losses. - Cosmic ray acceleration, Compton scattering: application to AGN & micro-quasar jets

Jets, beaming, superluminal motion. Cosmic rays, shock waves and Fermi acceleration. Comptonization by hot plasma.

### Study Resources

**Previous Qualifier Examinations:**

2015 Exam

2014 Exam

2013 Exam

2012 Exam

2011 Exam

2010 Exam

2001 – 2009 Exams