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Course, academic year 2018/2019
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Theory of nanosccopic systems I - NJSF132
Title in English: Teorie nanoskopických systémů I
Guaranteed by: Institute of Particle and Nuclear Physics (32-UCJF)
Faculty: Faculty of Mathematics and Physics
Actual: from 2015
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: prof. RNDr. Jan Kvasil, DrSc.
Annotation - Czech
Last update: T_UCJF (19.03.2015)
Modely nezávislých fermionů a bosonů Hartree-Fock teorie fermionů a bosonů (Gross-Pitajevského rovnice, HF metoda při konečné teplotě) Brueckner-Hartree-Fock teorie (G-matice pro 2D elektronový plyn) Hustotní (density) funkcionální teorie (DFT) (příklady aplikací DFT – Thomas-Fermi teorie atomu, základní stav rozpuštěného plynu bosonů, Kohn-Sham rovnice) Kvantové body v magnetickém poli (model nezávislých částic pro kvantové body, Hallův jev, spintronika) Monte Carlo metody
Course completion requirements - Czech
Last update: doc. Mgr. Milan Krtička, Ph.D. (10.06.2019)

Složení ústní zkoušky.

Literature -
Last update: T_UCJF (19.03.2015)

Lipparini E., Modern Many-Particle Physics - Atomic Gases, Quantum Dots, Quantum Fluids, World Scientific Co., Singapore, 2003

Imry Y., Introduction to Mesoscopic Physics, Oxford University Press, Oxford, 1997

Rammer J., Quantum Transport Theory, Perseum Books, Reading, Massachusetts, 1998

Requirements to the exam - Czech
Last update: doc. Mgr. Milan Krtička, Ph.D. (10.06.2019)

Požadavky ke zkoušce odpovídají sylabu předmětu v rozsahu prezentovaném na přednášce.

Syllabus -
Last update: T_UCJF (21.05.2008)
1. Independent fermion and boson models
bosons, fermions, one- and two-body operators, density matrix, ideal Bose gas confined in a harmonic potential, Fermi gas (excited states, polarized Fermi gas), finite temperature and quasiparticles;

2. Hartree-Fock (HF) theory for fermions and bosons
HF method for fermions (example of physical systems of fermions treated by HF method, example of infinite systems treated by HF method), HF method for bosons, Gross-Pitaevski equations, HF method in the second quantization language, HF at finite temperature, Hartree-Fock-Bogoliubov and BCS;

3. Brueckner-Hartree-Fock (BHF) theory
Lippman-Schwinger equation, Bethe-Goldstone equation, one-dimensional fermion systems (numerical results for different systems), g-matrix for the 2D electron gas (decomposition in partial waves, separable approximation, g-matrix expansion, numerical results);

4. Density functional theory (DFT)
Density functional formalism, examples of application of the DFT (Thomas-Fermi theory of atom, the Gross-Pitaevski theory for ground state of diluted gas of bosons), Kohn-Sham equation, the Local Density Approximation (LDA) for the exchange-correlation energy, the Local Spin Density Approximation (LSDA), inclusion of current terms in the DFT (CDFT), ensemble density functional theory, DFT for strongly correlated systems (nuclei and helium), DFT for mixed systems, symmetries and mean field theories;

5. Quantum dots in a magnetic field
Independent particle model for quantum dots ( case, case, the maximum density droplet state), fractional regime, Hall effect, elliptical quantum dots (analogies with the Bose-Einstein condensate in rotating trap), spin-orbit coupling and spintronics, the DFT for quantum dots in a magnetic field, the Aharonov-Bohm effect and quantum rings);

6. Monte Carlo methods
Standard quadrature formulae, random variable distribution and central limit theorem, calculation of integrals by Monte Carlo method, Markov chain, the metropolis algorithm, variational Monte Carlo methods and quantum mechanics, propagation of a state in imaginary time, Schrodinger equation in imaginary time, importance sampling, fermion systems and the sign problem.

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