Experiments conducted near the turn of the twentieth century indicated that the constituents of ordinary matter, the atoms, have themselves internal structure. In particular, they seemed to be composed of essentially point-like, charged particles. Why is it, then, that atoms have definite sizes? Why do the negatively-charged electrons not spiral in and coalesce onto the positively-charged nuclei, as they could in classical Newtonian mechanics? Given that most of "one atom" seems to be "empty space", why is it that two atoms seem to take up about twice the space of one and that they do not overlap in a significant way? Why are ordinary matter "stable"?
That answers to these question can be obtained, through a series of rigorous mathematical arguments, starting from Schrödinger's equation, is one of the great successes of quantum mechanics, and the subject of this course. We will start by formulating quantum mechanics mathematically, which allows us to cast the stability problem as one of partial differential equations and functional analysis. The main mathematical portion of the course has two parts. First is the development of the Lieb-Thirring type inequalities for Schrödinger operators; these inequalities provide spectral information on the operators, and have broad applications beyond the stability-of-matter problem. Second is an analysis of properties of the Coulomb/Newton potential. The course will conclude with some applications to mathematical physics, with specific emphasis on the question of "stability of matter".
E. Lieb and R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University Press : We will aim to cover chapters 1-7 of this book. Time permitting additional applications from the latter chapters will also be discussed.
E. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society GSM14 : mostly as a reference for additional background material in analysis.
MTH828 (Real Analysis) and MTH849 (PDE II) are strongly recommended. Knowledge of basic classical and quantum mechanics are not requited, but can be helpful
Other syllabus information
Course meets Tuesdays and Thursdays, 9:30am - 12:20pm, in Wells Hall A106.
Summer session runs from May 14 through June 28.
Time permitting there may be opportunities for student presentations.