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Stability of Matter in Quantum Mechanics

Course Type: 
PhD Course
Academic Year: 
October - January
50 h
  • 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:


  1. Introduction to quantum mechanics. The hydrogenic atom. Uncertainty principles, stability of matter of the first kind.
  2. Many particle systems, bosons and fermions, density matrices. Introduction to large Coulomb systems, as models for atoms and molecules.
  3. Lieb-Thirring inequalities, semiclassical approximations.
  4. Thomas-Fermi theory. The TF energy functional. Existence and uniqueness of the minimizer. The no-binding theorem.
  5. Proof of stability of matter via TF theory.
  6. Derivation of TF theory for large quantum systems.
  7. Quasi-free states, Hartree-Fock theory and the correlation energy.
  8. 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: 


  1. E. H. Lieb and R. Seiringer. Stability of Matter. Cambridge University Press.
  2. E. H. Lieb and M. Loss. Analysis. American Mathematical Society.
  3. E. H. Lieb. Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603-641 (1981).
  4. N. Benedikter, M. Porta and B. Schlein. Effective evolution equations from quantum dynamics. SpringerBriefs in Mathematical Physics 7, (2016).


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