Fachbereich Mathematik

# Mathematical Quantum Theory

SUMMARY OF LECTURES

Lecture 1. Introduction to classical mechanics, Hamiltonian evolution. The wave function and the Schrödinger equation. Properties of Lp spaces, Hilbert spaces.

Lecture 2. Postulates of quantum mechanics. Observables and their quantum averages. Heisenberg's uncertainty principle. Solution of the Schrödinger equation in finite dimensional Hilbert spaces. Measurements in quantum mechanics.

Lecture 3. (by G. Marcelli) An example of quantum mechanical measurement process: the Stern-Gerlach experiment.

Lecture 4. The Fourier transform of L1 functions. The Schwartz space. Definition of metric on the Schwartz space. Completeness.

Lecture 5. Properties of the Fourier transform on the Schwartz space. Continuity of the Fourier map on S, bijectivity. Plancherel theorem.

Lecture 6. Proof of Plancherel theorem. Extension of the Fourier transform to L2. Existence and uniqueness of the solution of the free Schroedinger equation on the Schwartz space.

Lecture 7. Behavior of the solution of the free Schroedinger equation. Free propagator, pseudodifferential operators. Comparison of Schroedinger and heat equation. Probability distribution of the asymptotic momentum for the solution of the free Schroedinger equation. Momentum operator.

Lecture 8. Tempered distributions. Weak and weak* convergence. Examples: identification of functions with distributions. Delta distribution. Fourier transform of a distributions. Distributional derivative. Weak solutions of the free Schroedinger equation. Existence and uniqueness of the solution on the space of tempered distributions.

Lecture 9. Strong solutions of the free Schroedinger equation. Sobolev spaces. Properties; completeneness, norm, scalar product. Bounded linear operators. Notions of convergence for operators. The free propagator is strongly continuous but not norm continuous. Strongly continuous 1-parameter unitary groups. Generators.

Lecture 10. Properties of generators. Another example of quantum Hamiltonian: the harmonic oscillator.

Lecture 11. Eigenstates of the one-dimensional harmonic oscillator. Ground state, Simplicity of the eigenvalues. Relation with the Heisenberg uncertainty principle. Orthogonality. Representation in terms of Hermite polynomials.

Lecture 12. Review of Hilbert spaces. Separable spaces, orthonormal bases. Parseval's identity, isometry with the \ell^{2} space. Strong solutions of the time-dependent Schroedinger equation associated to the harmonic oscillator. Energy conservation.

Lecture 13. Qualitative properties of the dynamics generated by the Hamiltonian of the harmonic oscillator. Confinement as a consequence of energy conservation. Periodicity. Riesz representation theorem, orthogonal splitting of Hilbert spaces. Adjoint of a bounded linear operator. Properties of the adjoint.

Lecture 14. Bounded self-adjoint operators generate unitary groups. Unbounded operators. Adjoint. Graph of an operator, closed operators. The adjoint of a densely defined operator is closed. Equivalent characterization of symmetric operators.

Lecture 15. Equivalent characterizations of self-adjoint operators. The spectrum of unbounded operators. Example: The spectrum of the Laplacian on H2.

Lecture 16. Spectrum of the adjoint operator. The spectrum of self-adjoint operators is real. Symmetric densely defined operators with real spectrum are self-adjoint. Self-adjointness of the harmonic oscillator. Neumann series, properties of the spectrum. Introduction to projection-valued measures.

Lecture 17. Introduction to the functional calculus. Projection-valued measures. Simple functions. Functional calculus for simple functions.

Lecture 18. Properties of the functional calculus for simple functions: adjoint, multiplication, addition. Extension to bounded measurable functions. Extension to unbounded measurable functions.

Lecture 19. Functional calculus for unbounded measurable functions. Domain. Herglotz functions.

Lecture 20. The resolvent of a self-adjoint operator is a Herglotz function. Herglotz theorem.

Lecture 21. Construction of the projection valued measure. Examples: harmonic oscillator, free Laplacian. Physical interpretation of the spectral theorem.

Lecture 22. Existence and uniqueness of the solution of the time-dependent Schroedinger equation. Lebesgue theorem. Spectral subspaces. Decomposition of the spectrum.

Lecture 23. Dynamical interpretation of the spectral subspaces: RAGE theorem. Wiener theorem; proof of the first part of RAGE theorem.

Lecture 24. RAGE theorem, end of proof. Physical interpretation. Hydrogen atom, introduction.

Lecture 25. Kato-Rellich theorem. Application to the hydrogen atom: relative boundedness of the Coulomb potential with respect to the Laplacian.

Lecture 26. Example of non-selfadjoint Hamiltonian with real valued, unbounded potential. Coulomb uncertainty principle, and stability of the hydrogen atom. Ground state of the hydrogen atom. Discrete and essential spectrum, stability of the essential spectrum against relatively compact perturbations.

Lecture 27. Rotations in quantum mechanics. Angular momentum operator. Hamiltonians with radial potential commute with the angular momentum operator. Noether theorem in quantum mechanics. Commutator identities for the angular momentum operator. Raising and lowering operators. Eigenvalues of L_{3} and L^{2}.

Lecture 28. Joint eigenstates of L_{3} and L^{2} on the sphere. The hydrogen atom: solution of the radial equation in a given angular momentum sector. Excited states.

References:

E. H. Lieb, M. Loss. Analysis. AMS.

L. C. Evans. Partial Differential Equations. AMS.

M. Reed, B. Simon. Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press.

G. Teschl. Mathematical Methods in Quantum Mechanics. AMS.

J. J. Sakurai. Modern Quantum Mechanics. Prentice Hall.

L. D. Landau, L. M. Lifshitz. Quantum Mechanics: Non-relativistic theory. Pergamon Press.

B. Thaller. Advanced Visual Quantum Mechanics. Springer.