# V. Qualifying Examinations

Passing the written qualifying examination is a requirement for all PhD students. Students have up to three opportunities to pass the five sections of the written qualifying examination:

- Classical Mechanics
- Electricity & Magnetism
- Mathematical Methods of Physics
- Quantum Mechanics
- Statistical Mechanics

Each section is a three hour, closed-book written examination, with one sheet of information (formulae, equations, etc.) provided. Each test has three questions. Students are required to turn in solutions to only two of these. In order to pass a section, students need to score at least 12/20 points, where each question is worth a maximum of 10 points. It is not necessary for a student to retake any section which they complete successfully**.** **Students are required to take all unpassed sections of the exam each time the exam is offered or forfeit that attempt**.

UC Santa Cruz is committed to creating an academic environment that supports its diverse student body. If you are a student with a disability who requires accommodations for the Written Qualifying examinations, please submit your Accommodation Authorization Letter from the Disability Resource Center (DRC) to the Graduate Program Coordinator privately prior to the scheduled exams.

We encourage all students who may benefit from learning more about DRC services to contact DRC by phone at 831-459-2089 or by email at drcgrads@ucsc.edu.

All students in the Ph.D. program must pass a qualifying examination consisting of five written tests in the areas of Mathematical Methods for Physics, Classical Mechanics, Quantum Mechanics, Statistical Mechanics, and Electricity and Magnetism. Students have a first opportunity to take these five tests at the beginning of their first year. Once a student passes an examination in any one of the five areas they do not need to take an exam in that area again. If necessary, each student has a second opportunity to pass the written tests at the beginning of the second year. Students with at most one or two failed tests have a third opportunity to pass their remaining tests at the beginning of the winter quarter of their second year. Students who fail any of the remaining tests at this third and last attempt, and students who have not passed three or more of the five written tests after two attempts can either transfer to the terminal MS program (the MS degree is automatically awarded to students who passed at least ⅘ sections, and it requires an additional written research thesis for those who only passed ⅗ sections), or appeal to the Graduate Committee to continue on the PhD route. In this latter case, the Graduate Committee considers whether there is evidence of likely success in the PhD program. The Committee evaluates and reviews the student’s progress towards candidacy, including performance in courses and progress in research, and recommends possible remedial coursework or an oral examination, or recommends that the student transfer to the terminal MS route.

Past exams may be viewed at the password-protected webpage https://drive.google.com/drive

Textbooks covering material at the level of the examination include:

- Mechanics: Marion and Thornton,
*Classical Dynamics of Particles and Systems*, (all chapters). This is an undergraduate level text, used in many universities in the junior year. - Electricity and Magnetism: Jackson,
*Classical Electrodynamics*, esp. chapters 1-9, 11-14 (in principle, anything in the text may be covered in the exam). - Quantum Mechanics: Shankar,
*Quantum Mechanics*, entire text. One might want to supplement this with some text with a more extensive treatment of topics such as scattering theory and perturbation theory, with more physical examples. Possible texts include those of Sakurai and Baym. - Mathematical Methods of Physics: This is covered at an advanced undergraduate level. Suitable texts include those of Boas and of Arfken, in their entirety. This section tends to be somewhat unpredictable, but it is good to have a mastery of elementary differential equations and complex variables (particularly contour integration, and topics such as convergence of series, etc.). Topics like Laplace transforms, fourier analysis, and the like appear with some regularity. You won't be expected to remember long formulas about special functions, but you might be given information (e.g. recursion relations, integral representations) and be expected to manipulate it to derive results.
- Statistical Mechanics: Wannier,
*Statistical Mechanics*, and Plischke and Bergersen, "Equilibrium Statistical Physics." This section is covered at the graduate level although many problems can be solved with an undergraduate level of understanding.

Some advice on preparation:

- Allow plenty of time. The summer prior to the exam, you should try to minimize other commitments (research, travel, etc.). Plenty means a great deal -- the equivalent of many weeks, full time.
- Don't expect that you will master material simply by reading. It is important to work problems, from small exercises, to textbook problems and past exams. All studying should be done with pen and paper in hand. Invent exercises: derive formulas in texts, apply them to familiar day to day problems, etc. It is a good idea to spend some of your time studying with your friends. This will give you feedback on problems, and is also fun and good for your morale.
- Try to take a positive view of this experience. This is the last time you will be encouraged to take a broad view of physics. It is an opportunity to fill in gaps in your knowledge, explore material you didn't learn well in the past, read books that you have always wanted to read (e.g. some students study the Feynman lectures while preparing).

Statistical Mechanics Written Qualifying Exam Syllabus

*Note that the list of topic is only meant to provide a broad indication of the material covered by the written qualifying exam. This is not the syllabus for course Physics 219!*

Note that questions that build on simple extensions or applications of the topics listed below might also appear in the test.

- Thermodynamics
- The zeroth, first, and second law
- Carnot engines
- Entropy
- Equilibrium and thermodynamic potentials
- Stability conditions
- The third law

- Probability
- Random variables
- Probability distributions
- Many random variables
- Sums of random variables and the central limit theorem
- Rules for large numbers
- Information, Entropy, and Estimation

- Kinetic theory of gases
- Liouville’s theorem
- The Boltzmann equation

- Classical statistical mechanics
- The microcanonical ensemble
- Finite -level systems (2 , 3 state systems )
- The ideal gas
- Mixing entropy and the Gibbs paradox
- The canonical ensemble
- The Gibbs canonical ensemble
- The grand canonical ensemble
- Fluctuations in ensembles and relation to susceptibilities

- Quantum statistical mechanics

- Fermi and Bose Distributions
- Black-body radiation
- Hilbert space of identical particles
- Canonical formulation
- Grand canonical formulation
- Degenerate Fermi gas, Sommerfeld expansion
- Degenerate Bose gas, Bose condensation and Superfluid He
^{4}

- Interacting particles
- The cumulant expansion
- The cluster expansion
- Second virial coefficient and van der Waals equation
- Breakdown of the van der Waals equation
- Mean-field theory, Phase transitions (1
^{st}and 2^{nd }order), Critical behavior, Exponents

- Statistical fields
- The Landau theory of 2
^{nd}order phase transitions - Saddle point approximation and mean-field theory
- Discrete symmetry breaking and domain walls, Energy entropy arguments of Landau Lifshitz and Peierls domain wall entropy
- Exact solution of 1-d Ising model, Transfer matrix formulation

- The Landau theory of 2

- Fluctuations
- Scattering and fluctuations
- Correlation functions and susceptibilities
- Fluctuation corrections to the saddle point

Relevant recommended textbooks for SM:

- Mehran Kardar, Statistical Physics of Particles
- Mehran Kardar, Statistical Physics of Fields
- Pathria, Statistical Mechanics
- Kittel and Kroemer, Thermal Physics
- Plischke and Bergersen, Equilibrium Statistical Physics
- Wannier, Statistical Mechanics