SubjectsSubjects(version: 875)
Course, academic year 2020/2021
Selected Chapters on Mathematical Physics - NTMF025
Title: Vybrané kapitoly z matematické fyziky
Guaranteed by: Institute of Theoretical Physics (32-UTF)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Additional information:
Guarantor: prof. RNDr. Pavel Exner, DrSc.
Classification: Physics > Theoretical and Math. Physics
Comes under: Doporučené přednášky 1/2
Annotation -
Last update: T_UTF (22.05.2001)
Advanced parts of quantum theory: operators on Hilbert spaces; postulates of quantum mechanics, states and observables in quantum mechanics; global and local uncertainty relations; canonical commutation relations; time evolution, Schrödinger operators; point and contact interactions. For the 4th and 5th year of the TF and JSF studies and for doctoral students.
Course completion requirements - Czech
Last update: doc. RNDr. Karel Houfek, Ph.D. (11.06.2019)

Ústní zkouška

Literature - Czech
Last update: RNDr. Pavel Zakouřil, Ph.D. (05.08.2002)

J. Blank, P. Exner, M. Havlíček: Lineární operátory v kvantové fyzice, Karolinum, Praha 1993

Requirements to the exam - Czech
Last update: doc. RNDr. Karel Houfek, Ph.D. (11.06.2019)

Zkouška je ústní, požadavky odpovídají sylabu, v detailech pak tomu, co bylo během semestru odpřednášeno.

Syllabus -
Last update: T_UTF (13.05.2003)
States and observables in quantum mechanics.
A survey of properties of Hilbert spaces and operators on them. Spectral theorem and spectral types of self-adjoint operators, the theory of self-adjoint extensions. Basic postulates of quantum mechanics. Examples of simple quantum systems. Mixed states, superselection rules. Compatibility of observables. Algebraic formulation of the quantum theory.

Global and local uncertainty relations.
Heisenberg relations. Hilbert space of analytical functions. Coherent states. Local uncertainty relations.

Canonical commutation relations.
Nelson's example. Weyl relations: Stone - von Neumann theorem about the existence and uniqueness of their representation. Systems with an infinite number of degrees of freedom.

Time evolution.
The basic dynamical postulate. Time evolution pictures. Wave packet dispersion. Evolution of coherent states. Feynman "integrals". Time evolution of unstable systems. Friedrichs model.

Schroedinger operators.
Self-adjointness criteria. Discrete spectrum, its cardinality and structure. Essential spectrum, its stability. Systems with a boundary, quantum waveguides.

Point and contact interactions.
The one-dimensional case: definition of a point interaction, spectral and scattering properties. Kronig-Penney model. Point interactions in dimension two and three. Approximations by scaled potentials. Quantum mechanics on graphs.

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