Quantum Physics

Garant předmětu:

Igor Jex

Lecturer:

Michal Jex

Tutor:

Michal Jex

Supervisor:

Department of Physics

Synopsis:

The goal of the lecture is formulating and developing quantum theory as a physically motivated, but mathematically rigorous theory built upon the analysis of bounded and unbounded linear operators on separable Hilbert spaces. Previous knowledge of quantum mechanics is an advantage but not a predisposition for the course. The pivot point is the establishing of the main postulates of the theory and deriving their consequences for model systems, as well as a detailed study of the most commonly used observables in quantum mechanics. The lecture focuses on the exactness and proofs of the statements. Some common mistakes resulting from breaking the assumptions of these are also discussed.

Requirements:

Syllabus of lectures:

1. Advanced topics of spectral theory of closed operators: projector-valued measure and integration, essential and discrete spectrum, functions of operators, self-adjoint extensions of symmetric operators, multiplicative operators: connection between the function values, measure, and spectrum, spectral representation

2. Mathematical postulates of quantum mechanics: basic definitions, description of system state, observables, and measurement, measurement outcomes, post-measurement state, temporal evolution, composite systems, examples of common state spaces

3. Basic quantum observables: position and momentum operators in spatially bounded and unbounded systems, correspondence principle and Hamiltonian as the energy operator, domains and spectra, inequivalent self- adjoint extensions

4. Measurement compatibility, operator sets, simple spectrum, complete sets of commuting observables, state preparation. Worked examples of complete commuting sets – position, momentum, energy (relation with x-, p-, and E-representations), angular momentum and spin

5. Measuring incompatible observables, uncertainty relations, canonical commutation relations of position and momentum, Heisenberg relation, Weyl relations, Stone–von Neumann theorem

6. Spherically symmetric systems, angular momentum, spherical functions, bound and free states, vector and tensor operators, Wigner–Eckart theorem, angular momentum addition, orbital and spin angular momentum

7. Evolution of systems: unitary propagator, fundamental dynamical postulate, Stone theorem, Schrödinger equation, significance of Hamiltonian as the evolution generator: Ehrenfest theorem, conservative systems, stationary states, integrals of motion

8. Nonconservative systems, Schrödinger, Heisenberg, Dirac pictures of quantum mechanics, Heisenberg equation, Dyson series, Trotter formula, Feynman’s path integral

9. Description of composite systems, entanglement, reduced states – density matrix, ensembles of identical particles, second quantization: Fock space, physical vacuum, creation and annihilation operators, canonical commutation and anticommutation relations, time evolution, particle-particle interactions, basic notions and methods of scattering theory

10. Possible extensions of quantum mechanical postulates: superselection rules, mixed states, inseparable spaces, time-dependent observables, normal observables, positive-operator-valued measure, measurement effects

Syllabus of tutorials:

Practical examples of the usage of the concepts defined in the lecture, using these for solving simple problems, technical parts of proofs and special cases of some statements from the lecture, concrete physical systems: Bloch sphere, one- dimensional and isotropic harmonic oscillator, Coulomb potential, charged particle in electromagnetic field

Study Objective:

Knowledge:

To give graduates the basic quantum mechanics with a mathematically correct formulation.

Acquired skills:

Calculation of spektra of Hamiltonians and solution of other basic problems of quantum mechanics with rigorous mathematical methods.

Study materials:

Key references:

[1] G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators, American Mathematical Society, 2014

[2] J. Blank, P. Exner, M. Havlíček, Hilbert Space Operators in Quantum Physics, Springer, 2008

Recommended references:

[3] W. Thirring, Quantum Mathematical Physics: Atoms, Molecules and Large Systems, Springer Verlag, 2010

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• Matematická fyzika (compulsory course in the program)