After successful completion of the course, students are able to:
1) Identify relevant mathematical structures and concepts in condensed matter problems, e.g. eivenvectors, eigenvalues and other linear algebra structures, correlation functions, inital value and boundary value problem of differential equations, Fourier transform.
2) Use discrete and continuous symmetries to assess the correctness of numerical results and to simplify physical problems
3) Construct Fock space of fermionic problems and apply it to simple systems with few degrees of freedom
4) Use Fourier transforms both in space and time/frequency domain
5) Understand basic approximations use in condensed matter physics, e.g. strong and weak coupling expansion, large-N expansion, embedding, stochastic simulations
6) Calculate magnetic susceptibility of 6-site Hubbard molecule as a function of temperature
Foundations of quantum theory: causal evolution and measurement, observables, wavefunctions, Hermitean operators, evolution operator and Hamiltonian
Schrodinger equation, stationary states, time evolution, Hemitean and unitary operators
Basic statistical physics, Boltzmann factor
Symmetry and eigenvalue degeneracy
Quantum theory of many particles: fermions and bosons, Pauli principle, commutation relations, Pauli principle
Lattice models, translational symmetry
Exact diagonalization, construction of Fock space and fermionic operators
Correlation functions and linear response to external perturbations
Non-interacting electrons in periodic solids, Bloch theorem
There will be a pilot problem, to which we apply the studied concepts, a 6-atomic cyclic molecule.
