 # Mathematical Quantum Mechanics (Winter 2018-2019)

## News: Grades are available in the pin board in front of room B328. You can collect your final grade certificate from Mrs. Edith HĂ¶chst, office B330. Final exam and solutions. You can review your results on Friday Feb. 22 at room B004, 9:00-10:00. Lecture notes by Martin Peev Homework Sheets 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, 8:15-10:00, B004.
• Exercises: Monday, 16:15-18:00, B004.
• Tutorials: Friday, 14:15-16:00, B004.
• Additional tutorials: Tuesday, 16:15-18:00, B005.

References:
• S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd Ed., Springer, 2011.
• G. Teschl, Mathematical methods in quantum mechanics, AMS 2009.
• Reed and Simon, Methods of modern mathematical physics, Volume I-IV.
• Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext 2011
• Lieb-Loss: Analysis, Amer. Math. Soc. 2001.
• E. H. Lieb and R. Seiringer, The stability of matter in quantum mechanics, Cambridge University Press, 2009.

Exercises and Tutorials: There will be a homework sheet every week. You can handle your solutions in the exercise section, or put them in the exercise mail box (no. 31, first floor). 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: You can bring your notes (lecture notes, homework sheets and tutorial materials). Electronic devices are not allowed.

## Preliminary contents of the course

• Review of basic concepts/tools of analysis.
• Principles of quantum mechanics.
• Spectral theorem.
• Quantum dynamics.
• Scattering theory.
• An introduction to many-body quantum mechanics.
• Density functional theory.
• Quantum entropy.
• Stability/Instability of matter.

## Contents of the lectures

16.10.2018. Chapter 1: Review of analysis. Measure theory: measurable sets, measurable functions, Lebesgue integration, Monotone Convergence, Dominated Convergence, Fatou's lemma, Brezis-Lieb refinement of Fatou's lemma, Approximation of integrable functions by continuous functions with compact support; L^p spaces: definition of L^p norm, completeness of the norm (L^p spaces are Banach spaces), HĂ¶lder's inequality, dual space of L^p.

19.10.2018. L^p spaces (continued): weak convergence, Banach-Alaoglu theorem (weak compactness of bounded sequences), Banach-Steinhaus theorem (Uniform bounded principle, without proof). Convolution, Young inequality, approximation by convolution.

23.10.2018. Hardy-Littlewood-Sobolev inequality. Fourier transform: Plancherel theorem, inverse transform, Fourier transform of convolution, Fourier transform of derivatives. Sobolev space H^m(R^d). Hilbert space: orthogonality, Parseval's identity, Riesz representation theorem, weak convergence.

26.10.2018. Chapter 2: Principles of quantum mechanics. Postulates of quantum mechanics: states, observables, measurement, dynamics. Why do we need quantum mechanics? Strange observations and non-commutativity of observables, Einstein-Podolskyâ€“Rosen (EPR) paradox, Bell's inequality. Formal similarities of classical mechanics. Here is the lecture notes.

30.10.2018. Mathematical formulation of quantum mechanics. Heisenberg's and Hardy's uncertainty principles. Proof of the stability of hydrogen atom using Hardy's inequality. Chapter 3: Sobolev spaces. Distribution theory: test functions and distributions, locally integrable functions are distributions, fundamental lemma of calculus of variations, weak (distributional) derivatives. Two equivalent definitions of Sobolev space H^m(R^d). Smooth functions with compact support is dense in H^m(R^d).

2.11.2018. Sobolev inequalities for H^1(R^d): Scaling argument, Fourier transform of 1/|x|^s, standard Sobolev inequality for d>=3 (proof using Hardy-Littlewood-Sobolev inequality), application to the stability of hydrogen atom, Sobolev inequality in low dimensions.

6.11.2018. Sobolev embedding theorem: weak convergence in H^1(R^d), heat kernel, H^1 weak-convergence implies L^p strong-convergence in bounded sets. Sobolev inequalities/embeddings for H^s. Green function of Laplacian, mean-value theorem for harmonic functions, Newton's theorem.

9.11.2018. Derivative of |f| and diamagnetic inequality. Application of Sobolev embedding theorem: existence of ground state for hydrogen atom. Chapter 4: Spectral theorem. Bounded operators, compact operators, adjoint of an operator. Spectral theorem for compact operators.

13.11.2018. Proof of spectral theorem for compact operators. Definition of resolvent and spectrum. Basic properties of spectrum of bounded self-adjoint operators. Continuous functional calculus for bounded self-adjoint operators.

16.11.2018. States and observables in C*-algebra abstract setting. Spectral properties of hermitian, unitary, projection, and positive operators. Gel'fand isomorphism.

20.11.2018. Riesz-Markov representation theorem. Spectral measure. Zorn's lemma. Spectral theorem for bounded self-adjoint operator (multiplication operator version). Bounded functional calculus. Spectral theorem for bounded normal operators.

23.11.2018. Gelfand isomorphism (continued), representations in Hilbert space, the GNS construction. Lecture notes.

27.11.2018. Unbounded operators: densely defined domain, extension, adjoint operator, symmetric operator, self-adjoint operator, resolvent and spectrum. Spectral theorem for unbounded self-adjoint operators (Multiplication operator version). Functional calculus.

30.11.2018. Chapter 5: Self-adjoint extensions. Closure method. Essentially self-adjoint operators. Kato-Rellich method. Applications to SchrĂ¶dinger operators.

4.12.2018. Operators bounded from below. Quadratic form. Friedrichs extension. Chapter 6: Quantum dynamics. Stone theorem (strong solution version).

7.12.2018. Symmetries and unitary evolution. Density matrix and entropy. Lecture notes.

11.12.2018. Stone theorem (weak solution and strongly continuous one-parameter unitary group). Three fundamental questions in quantum mechanics: self-adjointness, spectral properties and scattering properties.

14.12.2018. Chapter 7: Bound states. Discrete spectrum and essential spectrum. Weyl's criterion for spectrum. Perturbation by relatively compact operators. Application to Schroedinger operator.

18.12.2018: Midterm exam

21.12.2018: lecture moved to 11.1.2019

8.1.2019. Chapter 8: Scattering theory. Overview I: physical motivation, potential scattering, RAGE theorem, scattering operators, asymptotic completeness, stationary scattering theory, Lippmannâ€“Schwinger equation. Lecture notes.

11.1.2019. Bound states (continued): Min-max principle, existence of (in)finitely many bound states of SchrĂ¶dinger operators, exponential decay of bound states

15.1.2019. CLR inequality on the number of bound states. Scattering theory (continued): Space localization of bound states, kernel of free SchrĂ¶dinger dynamics, RAGE theorem for free SchrĂ¶dinger dynamics

18.1.2019. Proof of RAGE theorem in general case

22.1.2019. Scattering theory overview (2): Asymptotic completeness - guide through the proof by Enss, Coulomb scattering, S-matrix, cross-section. Lecture notes (with consistent signs for the wave-operators)

25.1.2019. Detailed proof of existence of wave operators and asymptotic completeness for short range interactions using Cook's method.

29.1.2019. Kernel equation of wave operators. Chapter 9: Many-body quantum theory. Tensor product and many-body Hilbert space, Kato theorem on self-adjointness, HVZ theorem on essential spectrum. An overview on Kato's work by B. Simon (Sections 7 and 13 are particularly relevant to what we discussed in the course).

1.2.2019. Zhislin theorem for existence of bound states of atoms. Particle statistics: bosons and fermions. Pauli exclusion principle. The ground state energy of non-interacting systems. Density functional theories.

5.2.2019. Chapter 10: Quantum entropy. Overview: states vs. density matrices, desired properties of entropy, a probabilistic argument for von Neumann entropy, sub-additivity of entropy. Lecture notes

8.2.2019. Proof of some basic properties of entropy: Klein inequality and application to relative entropy, Gibbs variational principle, subadditivity, strong subadditivity (SSA), monotonicity of relative entropy under partial traces. If you like the topic please check Lecture notes by E.H. Lieb and An Introductory Course by E. Carlen.