Syllabus
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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.
Annotation
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