| Program |
Tuesday 11:00-13:00, Seminar Room, 1st Floor
Thursday 11:00-13:00, Seminar Room, 1st Floor
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| Goal of the Course |
The present course, for graduate and advanced undergraduate students, develops the method of second quantization in terms of which models of interacting particles are been introduced and analyzed. This is followed by an extensive presentation of the theory of Green functions at finite temperature. Applications include, among other, basic aspects of the phenomena of superconductivity, superfluidity, and magnetism.
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| Syllabus |
Brief survey of quantum mechanics. Harmonic oscillators:
The space of one-particle states. Periodic boundary conditions and the thermodynamic limit. Examples of creation and annihilation operators: the one-dimensional harmonic oscillator, quantum theory of the harmonic crystal, phonons.
Second quantization:
Identical particles. The space of n-particle states and the Fock space. Creation and annihilation operators. The occupation number basis. Hamiltonian and other useful operators in second quantization.
Statistical mechanics:
Summary of basic definitions and relations. The ideal quantum gas. Bose-Einstein and Fermi-Dirac distributions. Wick's theorem.
Elementary applications:
Non-interacting fermions. The Hartree-Fock method for interacting fermions.Application to the homogeneous electron gas.
Superfluidity:
Non-interacting bosons. Weakly interacting bosons and the canonical transformation method of Bogoliubov. Landau's criterion for superfluidity.The interpretation of superfluidity in liquid helium.
Superconductivity:
Meissner effect. The effective electron-electron interaction via phonon exchange. Cooper pairs. Binding energy. The microscopic theory of Bardeen-Cooper-Schrieffer (BCS): effective Hamiltonian, canonical transformation, solution of the BCS gap equation.
Magnetic ordering and spin waves:
Exchange interaction. The Heisenberg Hamiltonian for the ferro- and antiferromagnet. Representations of the spin operators in terms of bosons: Schwinger and Holstein-Primakoff. The 1/S expansion. Quantum fluctuations. Spin waves.
Green functions:
Green functions at finite temperature: retarded, advanced, causal, and thermal. Analytic properties. Correlation functions, spectral weight function, Lehmann representation, spectral moments, sum rules. Kramers-Kronig dispersion relations. The fluctuation-dissipation theorem. Bogoliubov inequality. Linear response theory: Kubo formula. Generalized susceptibilities: compressibility, spin magnetic susceptibility, dielectric function,electrical conductivity. The equations of motion method. Self-energy and the Dyson equation. Perturbation expansion. Decoupling approximations. Applications to simple models.
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