Lecturer:
Course Type:
PhD Course
Academic Year:
2020-2021
Period:
October - January
Duration:
50 h
Description:
- The course will discuss functional analytic methods for quantum mechanics, with a focus on the rigorous derivation of effective theories for many-body quantum systems. Topics to be covered include:
- Introduction to quantum mechanics. The hydrogenic atom. Uncertainty principles, stability of matter of the first kind.
- Many particle systems, bosons and fermions, density matrices. Introduction to large Coulomb systems, as models for atoms and molecules.
- Lieb-Thirring inequalities, semiclassical approximations.
- Thomas-Fermi theory. The TF energy functional. Existence and uniqueness of the minimizer. The no-binding theorem.
- Proof of stability of matter via TF theory.
- Derivation of TF theory for large quantum systems.
- Quasi-free states, Hartree-Fock theory and the correlation energy.
- Many-body quantum dynamics. Derivation of nonlinear effective evolution equations, the case of fermionic systems. Mean field regime, time-depentent Hartree-Fock equation, Vlasov equation.
- References:
- E. H. Lieb and R. Seiringer. Stability of Matter. Cambridge University Press.
- E. H. Lieb and M. Loss. Analysis. American Mathematical Society.
- E. H. Lieb. Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603-641 (1981).
- N. Benedikter, M. Porta and B. Schlein. Effective evolution equations from quantum dynamics. SpringerBriefs in Mathematical Physics 7, (2016).
Research Group:
Location:
A-136
Location:
A-136 and Zoom, sign in to get the link