Modern physics represents an extension of the search for the range of applicability of classical mechanics. Most current physical intuition is ultimately based on the concepts from the classical mechanics. Our department’s course in advanced classical mechanics, PY 783, is an upper level classical mechanics course aimed at developing a deep understanding of important concepts that can be applied in most branches of physics. Thus, this course is not only an indispensable part of the physicist’s education but also an essential preparation for advanced work and research in physics. Formal aspects of Lagrange’s Equations, Hamilton’s Principle, Canonical Transformations, Poisson Brackets and Hamilton-Jacobi Theory are presented. Application to the specific problems such as the central force problem and rigid body motion are practiced extensively. Extensions to the theory of relativity, covariant formulation, symmetry properties, and group structure are also emphasized. Because the lectures do not simply recapitulate the textbook but make the class interactive, it is essential to attend every full-lecture and to participate in class discussions. Students are expected to come prepared, not just to transcribe, but to think and respond.
Text: To Be Announced
Electrodynamics, as summarized by Maxwell’s Equations, is arguably the most successful theory that has ever been formulated, incorporating not only mechanics but also special relativity and describing nonrelativistic and relativistic phenomena on length scales ranging from at least the subnuclear to the size of the universe. Maxwell’s Equations provide an excellent framework for developing a thorough understanding of Gauss’, Stokes’ and Green’s Theorems, which are basic vector-calculus relations that are used in essentially all branches of physics. Finally, practical applications often involve spatial and temporal averaging, which connects microscopic behavior where the physics actually occurs to the macroscopic phenomena that we usually observe, and thus provide important examples of this often-neglected topic.
Starting with the Lorentz force law and a statement of Maxwell’s Equations, in PY785 we address the statics limit, developing an understanding of electro- and magnetostatic phenomena; the use of the Gauss’ and Stokes’ theorems to establish continuity conditions at both real and virtual boundaries; Green’s Theorem to provide general solutions of Laplace’s and Poisson’s equations for both Dirichlet and Neumann boundary conditions; and averaging to connect microscopic and macroscopic properties and thus provide the physical basis for the macroscopic form of Maxwell’s Equations. Approaches to solving the Laplace and Poisson equations are thoroughly treated, including the method of images, Green functions, and the use of series expansions and associated orthogonal functions in Cartesian, cylindrical, and spherical co-ordinates.
In PY786 we treat the time-dependent case, covering in addition some aspects of physical optics. Topics discussed include conservation laws, the Poynting vector with particular emphasis on implications for coherent and incoherent systems; plane-wave propagation in anisotropic as well as isotropic media; waveguides and resonant cavities, including optical fibers and lasers; radiation, scattering, and diffraction; and special relativity. To simplify what can easily become a confusing and difficult topic, we return regularly to the 10 fundamental relations: the four Maxwell equations, the three mathematical theorems listed above, the two constitutive relations, and averaging. Emphasis is placed on fundamentals and on setting up problems, noting that during their careers most students who take this course will be addressing problems that have not been solved previously and hence problems for which answers cannot be found in textbooks.
Text: Classical Electrodynamics by Jackson
PY 781 Intermediate Quantum Mechanics I
The development of quantum mechanics during the first thirty years of the 20th century has completely revolutionized Physics and our understanding of the laws of nature. While the results of quantum mechanics often seem counterintuitive (as compared to the more familiar classical mechanics), the predictive power of quantum mechanics in understanding the world at atomic length scales (and lower) has never failed. This course aims to develop a solid and thorough grounding in the basics of quantum mechanics beginning with a treatment of operator algebra, measurements, observables and the uncertainty relations. This will be followed by the development of nonrelativistic quantum theory which finds its expression in the form of Schroedinger’s equation, propagators and Feynmann Path Integrals. A discussion of the quantum theory of angular momentum, including the development of Clebsch-Gordon coefficients, Bell’s inequality and Tensor operators completes this basic treatment.
PY 782 Intermediate Quantum Mechanics II
In its simplest form, nonrelativistic quantum mechanics finds its expression in terms of the Schroedinger’s equation which, in most cases, cannot be solved exactly. The initial focus is therefore on a thorough treatment of approximate methods to solving quantum mechanical problems including the variational approach, time-independent and time-dependent perturbation theory. These methods will be illustrated with a number of classic problems in atomic physics including the Stark and Zeeman effects, and a semiclassical treatment of the radiation field. Other important topics to be covered include the treatment of many identical quantum particles, the subsequent development of the Hartree-Fock theory, bonding, and quantum scattering theory.
Text: Modern Quantum Mechanics by Sakurai