Selected Chapters on Mathematical Physics - NTMF025
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Last update: T_UTF (22.05.2001)
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Last update: doc. RNDr. Karel Houfek, Ph.D. (11.06.2019)
Ústní zkouška |
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Last update: RNDr. Pavel Zakouřil, Ph.D. (05.08.2002)
J. Blank, P. Exner, M. Havlíček: Lineární operátory v kvantové fyzice, Karolinum, Praha 1993 |
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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. |
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Last update: T_UTF (13.05.2003)
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. |