PH4990 Advanced Theoretical Physics

A graduate-level introduction to the methods of theoretical physics. Beginning with complex variable methods as a tool for solving problems in physics, the Kramers-Kronig formulas are derived (connection between analyticity and causality in stable physical systems), together with the Hilbert transform and its role in defining the analytic signal and the extension of phasors to time-varying systems. The stationary phase approximation is derived as a method of treating the high-frequency behavior of oscillatory integrals, important for wave transmission and scattering.The method of Green functions for solving inhomogeneous differential equations is developed, with applications of Green functions as propagators affecting the response of physical systems to the influence of sources. An introduction to integral equations provides the foundation for scattering problems in physics, both electromagnetic and quantum. The essentials of tensor calculus aredeveloped, the language of the general theory of relativity (necessary for understanding the Global Positioning System and other physical systems) and of physical processes in anisotropic media.

Prerequisite

PH3991 or equivalent coverage of partial differential equations, special functions, and Fourier analysis.

Lecture Hours

4

Lab Hours

0

Course Learning Outcomes

After completing this course, students will:

• Master complex variable methods and their application in theoretical physics: Students will learn how to use complex analysis, including the Kramers-Kronig relations and Hilbert transforms, to solve problems in physics. They will also understand the extension of phasors to time-varying systems and their role in defining analytic signals.
• Apply advanced mathematical techniques, such as the stationary phase approximation and Green's functions, to physical systems: Students will derive and use the stationary phase approximation to analyze high-frequency behavior in oscillatory integrals. They will also develop an understanding of Green's functions, using them to solve inhomogeneous differential equations and study the response of physical systems to external sources.
• Understand and use tensor calculus in theoretical physics, with applications in relativity and anisotropic media: Students will gain a foundational understanding of tensor calculus and its essential role in the general theory of relativity. They will apply tensor methods to problems related to physical systems such as the Global Positioning System and processes in anisotropic media.