Enrollment into the program, familiarity with the content of Quantum mechanics course.
Advanced quantum mechanics
2025/2026
Programme:
Physics, Second Cycle
Orientation:
Technical physics and photonics
Year:
1 year
Semester:
first
Kind:
optional
ECTS:
7
Language:
slovenian
Course director:
Lecturer (contact person):
Hours per week – 1. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Prerequisites
Content (Syllabus outline)
Composite quantum systems: Tensor pruduct of Hilbert spaces, product states, entangled states, partial trace, density operator.
Systems of identical particles: Symmetrization postulate. Bosons and fermions. The Slater determinant.
The method of second quantization: Fock space. Creation and annihilation operators for bosons and fermions. CCR/CAR. Field operators. Momentum representation. The models of quantum systems of coupled fermions and bosons.
Wick theoremand examples.
Fermions: Fermi sphere, particles and holes, excited states. Pair correlation function. Ground state of fermi systems in Hartree-Fock approximation.
Electron gas: Electronic states with Coulomb repulsion. Energy of Fermi gas as a function of density.
Fermi systems in one dimension and spin chains: Wigner-Jordan transformation and exact solutions of spin chains.
Atoms and molecules: Thomas-Fermi model of atom. Hartree-Fock equations for atoms and molecules. Born-Oppenheimer approximation.
Bosons: Free bosons, pair correlation function. Coherent states. Dilute bose
gas: Bose-Einstein condensation. The Bogoliubov method and canonical transfromations for excited states. Quasi-particles. Superfluidity.
Quantum propagator: Spectral functions, Lehman representation, physical interpretation of spectral function. Thermal (Gibbs) states and fluctuation-dissipation theorem.
Perturbation expansion: time-dependent perturbation theory. Linearresponse, dynamical suscptibility. Feynman diagrams in coordinate and momentum space. Dyson equation.
Readings
F. Schwabl: Advanced Quantum Mechanics (Springer, 1999),
A. S. Davydov, Quantum Mechanics (Pergamon Press, 1970),
A. L. Fetter, J. D. Walecka, Quantum Theory of Many-Particle Physics (Mc Graw Hill, 1971),
M. Rosina, Višja kvantna mehanika (DMFA, 1995).
Objectives and competences
Mastering fundamental knowledge and theoretical methods for describing quantum systems with few or many degrees of freedom and application for description and analysis of real mesoscopic systms as quantum dots, quantum wires, thin layers etc.
Intended learning outcomes
Knowledge and understanding:
Fundamental theoretical descriptions to quantum systems of many particles.
Application:
The methods of advanced quantum mechanics are a basis for formulating the models of real physical systems and their theoretical and experimental treatment.
Reflection:
Example of application of the methods of advanced quantum mechanics and statistical physics for describing properties of materials and quantum many-body systems in condensed matter.
Transferable skills: Transfer between theoretical methods and understanding fundamental properties of quantum systems.
Learning and teaching methods
Lectures, seminar excercises, home work, tutorial.
Assessment
Written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)
Lecturer's references
Spin-spin correlations of entangled qubit pairs in the Bohm interpretation of quantum mechanics,
A. Ramšak, J. Phys. A: Math. Theor. 45, 115310 (2012).
Exact nonadiabatic holonomic transformations of spin-orbit qubits,
T. Čadež, J.H. Jefferson, and A. Ramšak, Phys. Rev. Lett. 112, 150402 (2014).
Open XXZ Spin Chain: Nonequilibrium Steady State and a Strict Bound on Ballistic Transport,
T. Prosen, Phys. Rev. Lett. 106, 217206 (2011).
Exact Nonequilibrium Steady State of an Open Hubbard Chain,
T. Prosen, Phys. Rev. Lett. 112, 030603 (2014).
