Degree Requirements
New System of Requirements
The Physics Department has updated the requirements for the Ph.D. degree. It is intended that the new requirements should be in place, at least on a provisional basis, starting in the Fall Semester of 2021. Students enrolled prior to Fall Semester of 2021 will be allowed to satisfy either requirement, or a hybrid as discussed in the Transition Period section below.
Course Requirements
To allow students the flexibility to choose courses that are related to their field of research, there is no set required course track for physics graduate students.
Students who do not have preparation in graduate-level physics courses are strongly recommended to take our core graduate-level courses. Students who successfully complete these classes will be deemed to have satisfied the Qualifying Exam requirement.
In addition, there are the following requirements:
- All students must take three graduate-level (600-level or above) courses of three credits or more excluding the core graduate-level classes, seminars and independent studies. These classes need not be in the Physics Department.
- All students must take two semesters of a physics seminar course.
- All students must take the non-credit Foundations and Frontiers of Physics seminar. Typically, this is done during the second semester of graduate study. This non-credit seminar requirement is in addition to the two semesters of physics seminars.
Advancing to Candidacy
To advance to candidacy, all students must:
- Satisfy the Qualifying Examination requirement This can be done by either taking and passing the two qualifying exams or via successful completion of core graduate-level classes or by a combination. Details can be found here.
- Give a Preliminary Research Presentationwithin a year after passing the Qualifying Exam.
- Submit a Scholarly Paper within four years of entering the graduate program.
Scholarly Paper Requirement:
The purpose of the scholarly paper is for the student to demonstrate an understanding of their subfield sufficient for Ph.D. level research andto demonstrate competence in scientific writing. Accordingly, it should be written by the student without coauthors. The scholarly paper may include one or more of the following:
- a description of a completed original research project by the student
- summarize the state of the art of a subfield
- describe a proposed future research project by the student
- give a progress report on a partially completed research project
- describe the results of research completed by the student with collaborators including their research advisor; in such cases, the scholarly paper will differ from the version submitted for publication since the published will have coauthors
- Please note: the initial draft of a paper or a portion of such a paper can serve as part or all of the scholarly paper provided it is written solely by the student.
Additional Requirements for the Ph.D. Degree
To graduate with the Doctor of Philosophy degree, all students must:
- Complete all of the requirements for advancing to candidacy.
- Take at least 12 credits (2 semesters) of Doctoral Dissertation Research (PHYS899). Students who have reached candidacy will be registered for 6 credits of PHYS899 each semester until he/she graduates.
- Register for classes each Fall and Spring semester until graduation.
- Maintain a GPA of 3.0 or above.
- Write and successfully defend an original dissertation.
- Meet all degree deadlines, outlined here.
Transition Period
(For students enrolled prior to Fall 2021)
The old requirements differ from the new ones in two ways: The qualifier examination requirement and the course requirements. Students who enrolled in the program prior to the Fall 2021 semester have options of satisfying the new requirements or the old requirements or a hybrid. Moreover, since the old and new qualifier requirements do not match neatly, special accommodations may be made for students who enrolled prior to the Fall 2021 semester and have not yet satisfied the qualifier requirement .
The old requirements are listed below:
- All students must take two courses outside his/her area of specialization. One of these must be a 700 or 800 level physics course.
- Theory students must take either PHYS624 or PHYS625.
- All students must take two semesters of a physics seminar course.
- All students must take the non-credit Foundations and Frontiers of Physics seminar. Typically, this is done during the second semester of graduate study. This non-credit seminar requirement is in addition to the two semesters of physics seminars.
Core Classes
New System of Core Courses
Starting in Fall 2021, the Department of Physics is replacing our set core graduate-level physics courses.
Our new set of core courses is designed to reduce the time commitment by replacing the six courses of the old system with four courses. The intention is for more students to take the full set of classes than previously and thus to help ensure that students have a well-rounded basic graduate-level physics education. To further encourage students to take these classes—and take them seriously—an additional change is being instituted: Students who complete these courses successfully will be deemed to have satisfied the Ph.D. Qualifier Examination requirement.
The courses are designed to be at a level appropriate for incoming graduate students who have successfully completed a full undergraduate physics sequence including advanced undergraduate courses in classical mechanics, thermal and statistical physics and electricity & magnetism along with two semesters of quantum. Students are also expected to have mastered the mathematical methods relevant for those courses.
Students who have taken a number of graduate-level physics courses prior to joining our program may consider taking the Ph.D. Qualifier upon entering the program. If the Qualifier is passed, the student may skip this sequence of classes. However, we strongly discourage students who have not taken such graduate classes from pursuing this route.
Students whose undergraduate preparation has gaps may elect to postpone taking one or more of the core sequences until their second year in the program.
The Course Sequences
The new core courses are arranged in two two-semester sequences. One sequence—PHYS 610 (Fall) & PHYS 611 (Spring)—focuses on “Mathematical Methods and Their Applications in Classical Mechanics and Electrodynamics”. The other sequence—PHYS 612 (Fall) & PHYS 613 (Spring)—focuses on “Mathematical Methods and Their Applications in Classical Mechanics and Electrodynamics”.
Tentative Syllabus
Since these sequences are new, we anticipate that the courses as currently designed may change somewhat as we gain experience with them. However, the initial incarnation of these sequences as expected cover the following topics:
PHYS 610
PHYS 610
Course Title: Mathematical Methods and Their Applications in Classical Mechanics and Electrodynamics I
Description for Catalog: This course is the first course of a two-semester graduate level sequence on classical mechanics, electrodynamics and relativity and the mathematics that underlie these subjects. Mathematical methods will generally be introduced in the context of relevant physical problems.
Credits: 4
Topics/Syllabus:
- Calculus of variations and its application in classical dynamics.
- Introduction to calculus of variations
- Least action/ Lagrangian formulation of classical mechanics
- Invariance of formalism under change of variables; constraints
- very small number of simple examples
- cyclic variables, symmetry and conservation laws.
- Hamiltonian Formulation
- Liouville theorem, Poisson brackets
- Canonical transformations
- Action Angle variables
- Maxwell’s equations
- Maxwell’s equation in vacuum
- Charge conservation
- Energy density, Poynting vector and conservation of energy
- Scalar and vector potential, gauge transformations
- Maxwell’s equation in media
- Remind students of polarization, magnetization E, D, B&H
- Boundary conditions
- Wave propagation in media
- Dispersion relations, group and phase velocity
- Complex indices of refraction
- Orthogonal Function expansions and solutions to electro- and magneto-statics problems (2.5 weeks)
- Rectangular coordinates and Fourier decompositions
- Spherical coordinates and spherical harmonics
- Show but do not prove useful properties of spherical harmonics
- A very few simple examples
- Cylindrical coordinates and Bessel functions
- Show but do not prove useful properties of spherical harmonics
- A very few simple examples
- Complex Analysis and applications to wave propagation
- State but do not prove properties: Analytic functions and maps, branch cuts, contour integration, singularities, residue theorem, Taylor and Laurent series.
- Stationary phase/steepest descents
- Kramers-Kronig relation
Other topics as time permits.
PHYS 611
PHYS 611
Course Title: Mathematical Methods and Their Applications in Classical Mechanics and Electrodynamics II
Description for Catalog: This course is the second course of a two-semester graduate level sequence on classical mechanics, electrodynamics and relativity and the mathematics that underlie these subjects. Mathematical methods will generally be introduced in the context of relevant physical problems.
Credits: 4
Topics/Syllabus:
- Green’s functions and applications to mechanics and electrostatics
- General principle of Green’s functions
- Damped driven oscillator
- Simple application: electrostatics with a plane boundary via method of images for dielectric and/or conductors.
- Green’s function in spherical coordinates in free space
- Cartesian tensors and rotations in classical mechanics (highly simplified)
- Introduction to Cartesian tensors
- Rotations and Euler angles
- Moment of inertia tensor and simple examples of rigid body motion.
- Multipole expansions in electrostatics, magnetostatics
- Multipole expansion for static charge distributions
- Multipole expansion for static current distributions
- Radiation; multiple expansion in electrodynamics
- Retarded Green’s functions
- Fields from oscillating sources; multipole radiation
- radiation from moving charges (Lenard-Wiechart potentials
- Thomson scattering
- Relativity
- Four vector formulation as encoding time dilation, relativity of simultaneity etc.
- Four tensors, the field strength tensor and the covariance of Maxwell’s equations.
- Other topics as time permits.
PHYS 612
PHYS 612
Course Title: Quantum and Statistical Physics I
Description for Catalog: This course is the first course of a two-semester graduate level sequence on topics in quantum mechanics and statistical mechanics.
Credits: 4
Topics/Syllabus:
- Brief introduction/review of formalism of quantum mechanics
- Quantum states as vectors in a Hilbert space, linear algebra, bra & ket vectors, unitary and Hermitian operators, basis and change of basis, Dirac-delta function
- Postulates of QM (superposition, time evolution as a Unitary operator, Born rule), (non)commuting observables and uncertainty. canonical quantization (from classical to quantum), entanglement
- Elementary applications of Quantum Mechanics (mostly a rapid review)
- Two level system
- Free particle
- 1-d harmonic via raising and lower operator
- Tunneling and resonance in 1d
- Constant mag field Landau level
- Quantum mechanics of identical particles
- Foundations of Statistical Physics
- Goals of stat. mech./thermodynamics
- foundations of classical statistical mechanics.
- Foundations of quantum statistical mechanics Mixed states and density matrices
- Ensembles: microcanonical, canonical and grand canonical
- Thermo: thermodynamic potentials
- Brief treatment of Legendre transformations, minimization principles and the relevant thermodynamic potentials.
- Thermo: Carnot's theorem
- Ideal gases
- classical
- Ideal gases: bosonic ideal gasses and Bose-Einstein condensation
- Ideal gases: fermionic, degeneracy pressure
- Equipartition theorem
- gases with internal degrees of freedom, phonons, Einstein/Debye solids
- Phase transitions: Van der Waals, 1st order, critical points, mean field theory
- lattice models, mean field, intro. to universality.
Other topics as time permits.
PHYS 613
PHYS 613
Course Title: Quantum and Statistical Physics II
Description for Catalog: This course is the second course of a two-semester graduate level sequence on topics in quantum mechanics and statistical mechanics.
Credits: 4
Topics/Syllabus:
- Quantum mechanics of rotation and angular momenta
- spin ½: Pauli spinors, finite rotations
- Spherical potential for spinless particles, Spherical harmonics to solve
- Addition of angular momenta, Clebsch-Gordan coefficients
- Symmetries in quantum mechanics
- Symmetries, conservation laws and degeneracy.
- Continuous symmetries
- Translations/ momentum conservation
- Rotations/angular momentum
- Discrete symmetries
- Parity
- Discrete translations, Bloch’s theorem
- Time-independent Perturbation Theory
- formal expansion
- nondegenerate
- Examples (only a few)
- Feynman-Hellmann theorem
- degenerate
- Time-dependent Perturbation Theory
- interaction picture, formal development
- Fermi’s Golden Rule
- Sudden approximation; Adiabatic approximation
- Berry Phase.
- Born-Oppenheimer approximation
- Scattering
- Formal aspects of scattering theory
- Born Series, Born approximation
- partial wave expansion
- optical theorem
- bound states as poles in the T-matrix
- Other Topics as time permits.