# Graduate Qualifying Exam

**Exam format** There will be *three separate* exams: Classical Mechanics (CM), Electricity and Magnetism (EM), and Quantum Mechanics (QM), with 5 problems in each exam. *Passing condition for each exam: *60% minimum grade. All problems are at the level of WVU upper division undergraduate courses in CM, EM, and QM. Exam problems are designed to be solved in less than 30 minutes. The duration of each exam is 3 hours. The trio of exams is given twice per year, no later than the second week of fall and spring semesters, within a single seven day period [for example: CM on Monday, EM on Wednesday, QM on Friday].

*All entering students are required to take all three exams at the beginning of, or immediately prior to, their first semester.* The results of the initial exams may be used by the Graduate Studies committee to recommend students take upper-level undergraduate Physics courses. Following the initial exam, as needed, students can retake exams **up to three times,** *but no later than the beginning of their fourth semester of graduate studies.* For example: A student entering in Fall 2016 must take the exam in Fall 2016, and, if needed, he/she can retake some or all of the exams in Spring 2017, Fall 2017, and Spring 2018, i.e., their fourth semester. To retake the exam, students are required to be in good standing. The Graduate Studies committee will be empowered to make all decisions regarding any exceptions to the above deadlines.

**PhD Qualifier Committee:** There will be a single committee with a chair who guides three subcommittees for the CM, EM, and QM exams. Each subcommittee has typically five members who are in charge of the exams both in fall and in spring. Interactions between different subcommittees are guided by the committee chair. Committee members should typically serve three years to allow rotation through the department, with overlapping terms in place.

**Student guidance and the method of developing problems:** The exam is conceptually tied to WVU advanced undergraduate courses in CM, EM, and QM so exam makers and takers are guided by the syllabi in these courses. The textbooks and homework collections developed for these courses are used to help the students who do not pass some exams upon entry. The exams and homework assignment problems given in the WVU courses in CM/QM/EM are used by exam makers as an inspiration for the content of the PhD qualifying exams. Repeat problems are not required. Lists of topics for all three exams are given at the end of this document. Exam makers will be required to ensure that their problems are each solvable in 30 min. Thus, during the exam preparation, each problem maker will provide the committee chair with handwritten solution (key) of his/her problem, which will be no more than 2-3 pages long yet detailed enough to fairly reflect the realistic time needed to solve the problem. The solution should be written using the style of a typical exam taker who is knowledgeable enough to pass the exam.

**Transition period:** The new exams can be directly used as CM/EM/QM qualifiers for currently enrolled students who have not passed the current (old) form of CM/EM/QM qualifiers by the Summer of 2016. The deadlines for passing the exams are those which were in effect during the semester a student began graduate studies in the department.

**Grading Guide:** The following is intended as a guide to the committee members for their use in grading problems. It should also be of interest to the students to understand the overall grading philosophy. These guidelines will be carried out on all parts of a problem individually, so it pays to write something down for all parts of problems even if you don’t have time to write a complete answer. A good strategy in such cases is to *thoughtfully* explain how you might have attempted the problem, given more time. Avoid the temptation to write down “core dumps” of equations that might be vaguely related to the problem.

A(80-100%): Everything correct or nearly correct. The student masters both physics concepts and math methods used in physics. Only some minor imperfections.

B(60-80%): The student seems to master physics concepts: He/she correctly constructs equations needed to solve the problem, but misses some minor physics concepts or fails to complete math part of the problem entirely correctly. Yet, the student work does not contain any serious physics or math errors or flaws that would disqualify the student from earning PhD in exact sciences such as physics.

C(40-60%): The student correctly writes down the key physics equations needed to solve the problem, yet he/she does not demonstrate a complete understanding of the physics or the math skills really needed to solve the problem. The student work does contain some serious physics or math errors or flaws that would disqualify the student from earning PhD in exact sciences such as physics.

D(20-40%): The student shows that he/she knows some physics and math related to the problem, but not enough to answer this problem correctly. Misunderstanding or misapplying at least one major concept.

E(0-20%): The student shows that he/she does not understand the basic physics and math related to the problem.

**Study aids:** To aid students with their studying, solutions for previous exams are available below. An accompanying document “An Expert’s Approach to Solving Physics Problems” on how to approach problem solving and how problems are graded is available by clicking here. This document provides useful strategies for problem solving and critical thinking when tackling these problems, as illustrated by a complete discussion of an example problem. In addition, problems and solutions for the previous five years of qualifier exams are available in the department office for students to look at.

**Topics and textbooks:** Below is a list, for each exam, the current set of topics that examiners will use when setting questions. Also given are suggested undergraduate textbooks that the students can refer to when studying this material.

**Classical Mechanics Exam**

Suggested textbooks:

- “Classical Mechanics” by John R. Taylor, University Science Books, 2004;

- “Classical Dynamics of Particles and Systems” by Stephen T. Thornton and Jerry B. Marion, Brooks Cole, 2003;

- “University Physics with Modern Physics” (14th edition, 2016), by H.D. Young, R.A. Freedman, A. Lewis Ford.

Topics:

- Newton’s three laws of motion
- Inertial and non-inertial reference frames
- Projectile motion
- Energy and momentum of N body and continuous systems
- Simple harmonic oscillation
- Damped and/or driven oscillation
- Coupled oscillators and normal mode analysis
- Central force problems
- Rigid rotation
- Variational calculus (e.g. extremal paths on parameterized surfaces)
- Lagrangian and Hamiltonian equations of motion
- Collisions and scattering

**Electricity and Magnetism Exam**

Suggested textbooks:

- “University Physics with Modern Physics” (14th edition, 2016), by H.D. Young, R.A. Freedman, A. Lewis Ford.

- “Introduction to Electrodynamics” by David J. Griffiths, Third Edition, Prentice Hall, Upper Saddle River, New Jersey, 1999.

- “Electromagnetic Fields” by Ronald Wangsness, Second Edition, Wiley, 1986.

Topics:

- Vector and scalar potentials
- Coulomb’s law – point charges and continuous distributions Gauss’ law
- Conductors in electrostatic fields
- Electrostatic energy
- Laplace’s equation, boundary conditions
- Method of images, electric multipoles
- Electric fields in matter
- Electric field in conducting media
- Electric currents, Magnetic Fields, Lorentz force law
- The Biot-Savart law, Ampere’s law
- Magnetic fields in matter
- Magnetic Induction
- Faraday’s law
- Magnetic energy
- Magnetic multipoles
- Maxwell’s equations in general and isotropic homogeneous forms
- Plane waves in various medium; reflection and refraction
- Circuits and transmission lines
- Electromagnetic waves

**Quantum Mechanics Exam**

Suggested textbooks:

- “Quantum Mechanics” by David H. McIntyre, Pearson Addison-Wesley, 2012.
- “Introduction to Quantum Mechanics” by David J. Griffiths, Prentice-Hall, 1995.
- “Quantum Mechanics, Vol. I.” C. Cohen-Tannoudji et al.:, Wiley, 1992
- “Introduction to Quantum Mechanics”, R. Dicke and Whitke, Addison Wesley, 1960
- “Modern Physics”, Serway, Moses and Moyer, Brooks Cole, 2005

Topics:

- Interpretation of quantum mechanics, operator formalism
- The Generalized Uncertainty Principle
- The Bohr model
- Ehrenfest theorem
- Spin-1/2 and spin-1 particles
- Dirac and matrix representations of quantum mechanics
- Schrodinger equation and time-evolution: time-independent and time-dependent Hamiltonians
- Exact solutions to the 1-D Schrodinger equation
- 1D, 2D and 3D Quantum harmonic oscillator: ladder operator methods and coherent states
- Exact solutions to the cylindrical and spherical Schrodinger equation
- Exact solutions to the hydrogen atom Schrodinger equation
- Perturbation theory: nondegenerate and degenerate, up to second order in energy, first order in state corrections
- Angular momentum. Orbital and spin angular momentum
- addition of angular momentum
- Identical particles: fermions and bosons
- Perturbations on the hydrogen atom: fine structure, hyperfine structure, Zeeman effect
- Symmetries and conservation laws
- Variational Principle