 # Mathematical Quantum Mechanics (Winter 2019-2020)

## General Information

Goal: We study basic mathematical concepts of quantum mechanics.

Audience : Master students of Mathematics and Physics, TMP-Master. Bachelor students will get "Schein" if pass the course.

Time and place:
• Lectures: Tuesday and Friday, 8:15-9:45, B005.
• Exercises: Monday, 14:15-16:00, B005 and 16:15-18:00, B052.
• Tutorials: Wednesday 16:15-18:00, B045 and Friday 12:15-14:00, B005.

References:

Exercises and Tutorials: There will be a homework sheet every week. Doing the homework is the best way to learn the course's materials and to prepare for the final exam.

The tutorial section is provided to help you in reviewing the lectures. You can bring up your questions, discuss complementary materials, and try some extra exercises.

• You can get up to 100 points in the final exam.
• You can get up to 10 points in the midterm exam.
• You can get 1 point for every homework sheet if you solve at least 50% problems in the sheet.
You need 50 points to pass the course and 85 points to get the final grade 1.0.

Exams: We will have the midterm exam on December 20 (8:00-10:00) and the final exam on February 10 (10:00-13:00). You can bring your notes (lecture notes, homework sheets and tutorial materials). Electronic devices are not allowed.

## Preliminary contents of the course

• Principles of quantum mechanics.
• Basic tools of analysis.
• Self-adjoint operator and spectral theorem.
• Bound states.
• Quantum dynamics.
• Scattering theory.
• Resonances.
• Quantum entropy.
• Density functional theory.

## Contents of the lectures

15.10.2019. Chapter 1: Principles of quantum mechanics. Overview of mathematical concepts: wave functions, states, observables, Schroedinger equation. Hilbert spaces: inner product, orthogonality, basis, Parseval's identity, Riesz representation theorem, Banach-Alaoglu theorem, Banach-Steinhaus theorem. Bounded Operators: continuity/boundedness, adjoint operator, self-adjointness.

18.10.2019. C*-algebra of bounded operators. Algebraic formulation of quantum mechanics. Non-commutative property and unboundedness. Position-momentum commutator and Heisenberg uncertainty principle. Unbounded operators: dense domain, adjoint operator, self-adjointness, example of multiplication operator. Quick introduction to spectral theorem, Stone theorem for Schroedinger dynamics, semi-boundedness and Friedrichs' extension (without proofs). Hardy inequality and stability of hydrogen atoms.

22.10.2019. Postulates of quantum mechanics: states, observables, measurement, dynamics. Why do we need quantum mechanics? Einstein-Podolskyâ€“Rosen (EPR) paradox, Bell's inequality. Formal similarities of classical mechanics. Lecture notes.

25.10.2019. States and observables in C*-algebra abstract setting. Resolvent, openness of resolvent set. Spectral properties of hermitian, unitary, projection, and positive operators. Pure and mixed states. Representations, irreducibility.

29.10.2019. Chapter 2: Basic tools of analysis. Measure theory: measure space, sigma algebra, measurable functions, Lebesgue integration, Lebesgue Monotone/Dominated convergence theorems, Fatou's lemma, product measure and Fubini theorem. L^p space: Brezis-Lieb refinement of Fatou's lemma.

05.11.2019. L^p space (continued): HĂ¶lder's inequality, triangle inequality, completeness, dual space. Abstract functional analysis: Hahn-Banach theorem.

08.11.2019. Abstract functional analysis (continued): Banach-Steinhaus theorem (Uniform boundedness principle), Banach-Alaoglu theorem. Relations between strong/weak/pointwise convergences in L^p spaces.

12.11.2019. Real analysis: convolution, Young inequality, approximation by convolution, Fourier transform, kernel of operators, weak derivatives of L^2 functions via Fourier transform, Sobolev spaces H^m(R^d).

15.11.2019. Chapter 3: Spectral theorems. Spectral theorem for compact operators. Schatten norm, trace class operators, Hilbert-Schmidt operators. Continuous functional calculus. Spectral theorem for bounded self-adjoint operators (multiplication operator version).

19.11.2019. Characters and their properties, relation to pure states, weak*-topology, continuous functions on the character set and supremum norm, Gelfand isomorphism; GNS-construction: quotient space, pre-Hilbert space, representation of the algebra on the space. Lecture notes.

22.11.2019. Spectral theorem for bounded normal operators. Spectral theorem for unbounded self-adjoint operators.

26.11.2019. Chapter 4: Self-adjoint extensions. Closure of a symmetric operator. Operators bounded from below and quadratic forms. Lax-Milgram theorem on the correspondence between operators and quadratic form domains.

29.11.2019. Friedrichs extension for operators bounded from below. Kato-Rellich theorem on self-adjointness of perturbed operators. Further self-adjoint extensions by Cayley transform.

3.12.2019. Any representation of a C-star algebra is equivalent to a sum of GNS representations. Operational definition of symmetries and Wigner's theorem. Non-linear time-evolution violates the 2nd law of thermodynamics. Lecture notes.

6.12.2019. Chapter 5: Bound states. Weyl's Criterion for spectrum, discrete spectrum and essential spectrum. Relative compact perturbation.

10.12.2019. Min-max principle.

13.12.2019. Sobolev inequalities for H^s(R^d).

17.12.2019. Rellich-Kondrachov theorem. Application to SchrĂ¶dinger operators: self-adjointness and essential spectrum.

20.12.2019. Midterm exam.

7.1.2020. Scattering theory: concept of a scattering experiment, cross section. Potential scattering: Lippmann-Schwinger equation. Scattering operators: intuition and definitions of existence, uniqueness, and asymptotic completeness. Long term time-evolution and spectrum: RAGE theorem (stated and interpreted). Scattering theory is comparing two dynamics, intertwining property. Lecture notes.

10.1.2020. Number of bound states: finite or infinite? Existence of infinitely many bound states with long range potentials. Cwikel-Lieb-Rozenblum (CLR) bound. IMS localization formula. Exponential decay of eigenfunctions. SchrĂ¶dinger operators with trapping potentials.

14.1.2020. Chapter 6: Quantum dynamics. Time-dependent SchrĂ¶dinger equation. Existence and uniqueness of solution.

17.1.2020. Stone theorem on strongly continuous one-parameter unitary groups. Integral representation of free SchrĂ¶dinger dynamics. RAGE theorem for Laplacian.

21.1.2020. RAGE theorem on the dynamical characterization of spectral subspaces.

24.1.2020. Concepts and some ideas for the proof of asymptotic completeness: short range potentials, dilation operator, in- and out-spaces, the role of compactness and the use of spectral constraints, Perry's estimate. Failure of standard scattering theory for the Coulomb potential and physics reasons. Explicit form of the scattering operators (Abel limit). Relation to the Lippmann-Schwinger equation. Lecture notes.

28.1.2020. Wave operators. Cook's method. Proof of asymptotic completeness for short-range potentials.

31.1.2020. Chapter 7: Entropy. Quantum entropy and relative entropy. Klein inequality. Gibbs variational principle. Concavity of entropy. Lie-Trotter product formula. Golden-Thompson inequality. Frank-Lieb uncertainty principle on classical and quantum entropies.

4.2.2020. Density matrix as partial information about a quantum system. Normal states are density matrices. Counter-example of a non-normal state. Partial ordering of density matrices. Desired properties of entropy: monotonicity wrt. partial ordering, additivity. Von Neumann entropy follows from "strong additivity". Alternative motivation of von Neumann entropy by Boltzmann probability argument. (Lack of) continuity of entropy in absence of further constraints. Lecture notes.

7.2.2020. Tensor space. Partial trace. Sub-additivity of entropy. Araki purification lemma. Strong sub-additivity (SSA). Lieb's triple matrix inequality. Completely positive trace-preserving (CPT) maps / quantum channels. Stinespring characterization of CPT maps. Monotonicity of relative entropy / Data processing inequality.

10.2.2020. Final exam.