Mathematical physics is the academic consideration of topics at the intersection of math and physics. Those who are interested in a career researching and teaching about these subjects may consider earning a Doctor of Philosophy, or Ph.D. In addition to deciding upon the degree sought, students seeking a Ph.D. in this area should ensure that research facilities are available to further their interests, and that they can identify a faculty member who shares those research interests.
Ph.D. Program Options
As mathematical physics is an interdisciplinary field, some universities may offer the Ph.D. in physics; while in others it will be in mathematics or applied mathematics. Opportunities may also be available to earn a dual-major Ph.D.
One possible degree choice is in the area of applied mathematics. To pursue this degree, students will take courses in core mathematical areas, including topology and numerical analysis. They must also take a core course sequence in an outside area, in this case physics. Students will then design a plan of study with further advanced mathematics and physics courses. In addition to their coursework, students must pass qualifying and comprehensive examinations, and prepare and defend a dissertation. Often, universities will also offer seminars and workshops to review the research of university faculty. To apply to a Ph.D. in Applied Mathematics program, students should provide transcripts, GRE results, a statement of purpose and recommendations. Many students will enter these Ph.D. programs with a dual major in mathematics and science, or with a similar academic background that demonstrates exposure to both areas. Full financial support, through research and teaching assistantships, may be available depending on the university.
Earning a Ph.D. in Applied Physics is a second option for those interested in this field. Successful completion of the Ph.D. in Applied Physics typically begins with the completion of core physics classes, including classical mechanics, electromagnetism, and quantum mechanics. Students will also take coursework in the field of their minor; for students interested in mathematical physics this will be mathematics. Students then design further coursework with their advisor or a committee to best develop their research interests. In addition to courses, depending on the program, graduation requirements could include qualifying or area examinations, oral or final examinations in both physics and mathematics, and the completion of a dissertation that demonstrates independent research. Applicants should provide transcripts, GRE scores, a statement of purpose, and recommendations. Typically, undergraduate preparation will be in physics, engineering, or other related areas. Financial backing may be provided for those participating in research or teaching assistantships.
Program Information and Courses
Much of the work to be completed in a Ph.D. program will revolve around independent research. However, within the field of mathematical physics, there are some core courses to help students develop the tools needed for success in this discipline. Read on for some information about courses graduate students may encounter.
Partial Differential Equations
This course may aim to develop the problem solving skills necessary to utilize partial differential equations within a range of problems in physics. Specific topics may include convection, heat, and wave theories. The computations involved in solving these problems may be covered.
Advanced Perturbation Theory
A course in advanced perturbation theory may utilize and adapt various mathematical methods, including differential equations, partial differential equations, and integral equations. The application of these methods to a wide variety of problems in physics, including matched asymptotic expansions, period solutions, and harmonic resonance may be considered. Methods of averaging and approximation may be considered as well.
Digital Signal Processing
A course in digital signal processing may begin with the theoretical and mathematical tools to analyze and model various digital signal processing systems. Practical applications may then be considered. Students may utilize the tools obtained in their own independent research.
Quantum Information Processing
This course may begin by pursuing a theoretical introduction to the field of quantum computing and information. Then, the course may delve into real-world application of this theory, including cryptography, superdense coding, and teleportation. Students may engage in their own projects or research as a course component.
Mathematical Modeling in Physics
This course will explore a wide range of mathematical techniques that may be useful in physics research. Calculus, partial differential equations, and group theory will be used. Specific topics may include idealized fluids, acoustics, Navier-Stokes, Grassmann variables, and supersymmetry.
Within this course, students may come to understand how to use various computational techniques to research problems in physics. A variety of programming languages, including C++ and Java may be utilized, and a background in coding is preferred. Topics may include root finding, numerical integration, matrix inversion, and Monte Carlo techniques.
Mathematical physicists conduct research applying mathematical analysis to a range of problems in applied and theoretical physics. A Ph.D. in Applied Physics or Applied Mathematics can provide an entryway into this career.