# Qualifying Exam

All incoming graduate students are required to take a written "qualifying exam" at the start of their first semester at BYU, assuming it is offered. The exam tests their knowledge of fundamental undergraduate physics.

PhD students--and Master's students hoping to transition into the PhD program--are required to pass the exam at a threshold which has been about 60% in recent history. Those who do not pass it at first, and most do not, will have two more opportunities to pass within the next 20 months. If a student does not wish to take the exam his or her first semester, he or she will still only have two more chances to pass. The three attempts are typically (1) the first semester at BYU, (2) one year later in the fall and (3) the winter or spring following the second attempt. (You can find out from the Graduate Committee when exams are scheduled.) Any PhD student that does not pass this exam within 20 months of enrolling cannot continue in the PhD program but will be given the opportunity to apply their work toward an MS degree.

Master's students (those not interested in transitioning into the PhD program) must still take the exam, but are not required to pass it. It is used as a diagnostic to guide the student and committee in putting together an appropriate study list.

The exam consists of two parts: a two-hour multiple choice section and a four-hour worked-problem section. The two sections are typically given on separate days.

The multiple choice section has 88 questions. These are somewhat similar to the GRE Physics exam questions, but probably a bit easier on average. About half the questions are from the introductory physics sequence (physics 121, 123, 220, and 222 at BYU). The other half are from the more advanced undergraduate classes. It is not expected that students will know all the material, since it covers courses--such as acoustics and astronomy--that all may not have had. The exam score will therefore be calculated as a percentage out of 72 instead of out of 88, because it's expected that students will have a background in the material from at least 72 of the questions. (If a given student has a broader background than that, so much the better for him or her.)

The worked problem section consists of problems from each of 12 areas: mathematical physics, mechanics, thermodynamics, electricity and magnetism (I and II), quantum mechanics (I and II), optics, acoustics (I and II), and astronomy (I and II). Students choose 8 of the 12 problems to answer. Two exams from previous years are available so students can see examples of the type and difficulty level of problems:

In Fall 2010 the Graduate Committee asked people who had written one or more of these problems to put together a list of core conceptual and problem topics for their area, providing some specifics about which things may be "fair game" to be tested on. Here is the master list that was assembled:

**Mathematical Physics **(BYU
Phys 318)

(see textbook: Stanley J. Farlow, *Partial Differential Equations
for Scientists and Engineers*)

- Orthogonal functions (for example - Trigonometric, Bessel, Legendre, Fourier)
- Series expansion of an analytic function (in terms of these orthogonal functions). The Orthogonality conditions for these functions.
- Solve Partial differential equations infinite or semi-infinite domains
- Solve physical boundary value problems involving Cartesian, cylindrical, and spherical coordinates. This includes problems involving the wave equation and Laplace's equation.
- Use physical boundary conditions to generate the properly mathematical boundary conditions in boundary value and eigenvalue problems. This includes Dirichlet conditions, Neumann conditions and Robin conditions.
- Know how to use the principle of superposition to break up complicated boundary value problems (ones with multiple inhomogeneous boundary conditions) into simpler ones that can be solved.
- Eigenvalue problems
- Use Sturm-Liouville theory to generate and evaluate orthogonal functions for use in the solution of differential equations and expansion of functions.
- Separation of Variables
- Fourier transforms to expand functions on infinite or semi-infinite domains.

**Mechanics **(BYU
Phys 321)

(see textbook: J. Taylor, *Classical Mechanics*)

- Any mechanics problems at the Physics 121 level
- Linear and quadratic drag
- Potential energy in three dimensions: conservative forces, curl,
classical turning points

- The central force problem, centrifugal potential

- Conservation of momentum in three dimensions

- Damped and driven harmonic oscillators

- Rotational motion including moments and products of inertia,
diagonalizing the inertia tensor, Euler's equations

- Lagrangian techniques, finding differential equations and boundary
conditions

- Lagrange multipliers and forces of constraint

- Hamiltonian methods

- Coupled oscillators

- Accelerating reference frames

**Thermodynamics **(BYU
Phys 360)

(see textbook: Gould and Tobochnik, *Statistical and Thermal
Physics --*available online for free:
http://www.compadre.org/STP/.
See also Daniel Schroeder, *An Introduction to Thermal Physics*.)

- The laws of thermodynamics
- The concept of entropy
- The concept of free energy
- The equipartition theorem
- The ideal gas as a simple model
- 2-state systems as another model
- The Boltzmann factor
- Boltzmann statistics
- The partition function

**Electricity and Magnetism I **(BYU
Phys 441)

(see textbook: Griffith, *Introduction to
Electrodynamics*, 3rd edition, Chapters 1 to 6)

1. Coulomb force;
electric field (E); Gauss’s law and applications; boundary conditions;
Maxwell’s equations for E (static case).

2. Electric potential;
electrostatic energy; energy density.

3. Perfect conductors; Laplace’s
equation; separation of variables in 2d and 3d; method of images.

4.
Multipole expansion in electrostatics; dielectrics; polarization; electric
displacement field (D); electric susceptibility.

5. Lorentz force;
magnetic field (B); Maxwell’s equations for B (static case); Biot-Savart
law; boundary conditions; Ampere’s law and applications; vector potential;
Coulomb gauge .

6. Multipole expansion in magnetostatics; magnetization;
auxiliary field (H); magnetic materials; diamagnetism, paramagnetism;
magnetic susceptibility; ferromagnetism, hysteresis, domains.

**Electricity and Magnetism II** (BYU
Phys 442)

(see textbook: Griffith, *Introduction to
Electrodynamics*, 3rd edition, Chapters 7 to 12)

1. Electromotive
force; mutual and self inductance.

2. Maxwell's equations
incorporating matter; displacement current; boundary conditions.

3.
Continuity equation; Poynting's theorem; energy and momentum in
electromagnetic fields; wave equations for E and B.

4. Maxwell's
equations in terms of vector and scalar potentials; gauge choice ;
waveguides; retarded potentials; fields of a moving point charge.

5.
Radiation from dipoles and arbitrary sources; Larmor formula.

6.
Special relativity; Lorentz transformations; 4-vector velocity,
momentum, energy, current, potentials; Electromagnetic 4-tensor;
Maxwell's equations in 4-vector formulation.

**Quantum mechanics I ** (BYU
Phys 451)

(see textbook: McIntyre, *Quantum Mechanics*, Chapters 1 to
~9; previous text was D.J. Griffith, *Quantum Mechanics*, 2nd edition,
Chapters 1 to 5)

1. The Schrodinger equation

2. Probability, measurement outcomes, expectation values and standard
deviation

3. Heisenberg uncertainty principle

4. Time-Independent Schrodinger equation, and stationary states

5. Eigenstates and eigenvalues of an operator

6. Simple Harmonic Oscillator

7. Infinite square well and finite barrier potential

8. Free particle

9. Schrdinger equation in spherical coordinates

10. Hydrogen Atom

11. Angular Momentum and Spin

12 Two particles systems

**Quantum mechanics II ** (BYU
Phys 452)

(see textbook: McIntyre, *Quantum Mechanics*, Chapters ~10 to
16; previous text was D.J. Griffith, *Quantum Mechanics*, 2nd edition,
Chapters 6 to 11)

1. Non-Degenerate Perturbation Theory (formulae provided if necessary)

2. Degenerate Perturbation Theory (formulae provided if necessary)

3. Fine Structure of hydrogen atom

4. Zeeman Splitting

5. Variational Methods

6. WKB Approximation

7. Tunneling

8. Emission and absorption of radiation

9. Scattering and Born approximation

**Optics **(BYU
Phys 471)

(see textbook: Peatross and Ware,
*Physics of Light and
Optics*)

1. reflection/transmission at boundaries (Fresnel coefficients,
Brewsters angle, Snells law, etc.)

2. polarization effects (polarizers, waveplates, interfaces, Jones
Matrices)

3. ray optics and imaging (compound optical systems, ABCD matrices,
laser cavities)

4. Spatial and temporal coherence (Michelson interferometer, two slit
diffraction, coherence time/length)

5. Diffraction/Interference (scalar diffraction theory, Huygens
principal, Fresnel and Fraunhoffer approximations)

6. Wave propagation/dispersion using Fourier theory (group
delay/velocity, propagate a pulse through a dispersive medium, calculate
power spectrum)

**Acoustics I** (General Acoustics topics from Phys 123/318/other)

(see textbook: L.E. Kinsler, et al., *Fundamentals of Acoustics*,
4th edition)

1. Solutions of the one-dimensional and two-dimensional wave
equation, e.g., strings, pipes, membranes, with various boundary
conditions.

2. Wave interference and beating

3. Physical processes within the human ear and implications on human
hearing

4. Use of the decibel

5. Acoustic pressure and the inverse square law

6. Doppler shift

7. Concepts of outdoor sound propagation

8. Absorption and transmission phenomena

9. Addition of incoherent and coherent waves

10. Helmholtz resonators

**Acoustics II** (BYU
Phys 461)

(see textbook: L.E. Kinsler, et al., *Fundamentals of Acoustics*,
4th edition)

1. Definitions of and relationships between acoustic variables
(pressure, particle velocity, intensity, power)

2. Solution of the wave equation using complex variables in planar and
spherical coordinates

3. Definitions of impedance (mechanical, characteristic, specific
acoustic, radiation)

4. Using finite-impedance boundary conditions in wave equation solutions

5. Reflection and transmission of sound at fluid interfaces for normal
and oblique incidence

6. Transmission through and around single-layer panels and barriers

7. Properties that influence speed of sound in a medium

8. Modes in rectangular rooms

9. Simple sources of sound (monopoles, doublets, dipoles, line sources)

10. Operating principles and use of microphones and sound level meters

**Astronomy I **(BYU
Phys 427)

(see textbook: Carroll and Ostlie, *Modern
Astrophysics*; it is used as a reference book but not used
completely or exclusively)

1. Basic nomenclature of astronomical
objects

2. Literature and catalogs

3. Spherical coordinate systems
and spherical trigonometry

4. Time systems

5. Stellar positions
and motions, apparent and real

6. Measures of flux, luminosity and
brightness: magnitudes and color indices

7. Interstellar extinction

8. Stellar spectral types

9. The H-R Diagram

10. The stellar
luminosity function

11. Telescopes of all types

12. Spectrographs

13. Radiation detectors

14. Radiation and the Planck function

15.
The atomic velocity distribution and the Maxwellian distributions

16.
The atomic excitation distribution and the Boltzmann equation

17. The
atomic ionization distribution and the Saha equation

18. Electron
pressure and its dependence upon chemical composition, temperature and
total pressure

19. Molecular versus atomic spectroscopy, the
additional complications

(BYU **Astronomy 2**
Phys 428)

(see textbook: Carroll and Ostlie, *Modern
Astrophysics*; it is used as a reference book but not used
completely or exclusively)

1. Continuous spectra and sources of
continuous opacity

2. The spectrum of hydrogen

3. The spectra of
heavy elements: Grotrian diagrams, alkali metal spectra, Zeeman
effect

4. Synchrotron radiation

5. Radiative quantities and the
equation of radiative transfer

6. Spectral line broadening mechanisms
and their associated line profiles

7. Convolving line profiles

8.
Stellar spectra

9. The curve of growth, line strengths’ dependence
upon numbers of absorbers

10. Radial velocities

11. Binary stars