The course consists of 48 hours: 38 hours of theory and 10 hours of exercises. The main goal is to learn quantum phenomena of condensed matter physics by means of the path integral approach and analyzing quantum particles and quantum fields at finite temperature. The exam will be a colloquium at the blackboard with chalk, where the student will discuss two topics chosen on the spot by the examination board. The topics of the course are:
Path Integral for particles (14 hours)
The quantum propagator and its properties. Feynman path integral construction of the propagator. Propagator of the free particle. Saddle point and stationary phase approximation. Propagator of a particle in a harmonic potential. Wick rotation and thermodynamics. Quantum tunneling.
Path integral for bosons (18 hours)
Partition function of interacting bosons in quantum field theory. Bosonic coherent states. Semiclassical approximation and imaginary time. Coherent states and harmonic potential. Bosonic Matsubara frequencies. Ideal Bose gas of photons. Ideal Bose gas of massive particles. Interacting Bose gas: Bogoliubov spectrum. Dimensional regularization of Gaussian fluctuations. Gross-Pitaevskii equation and superfluid hydrodynamics. Landau criterion and superfluid fraction. Bright and dark solitons. Quantized vortices.
Path integral for fermions (16 hours)
Partition Function of interacting fermions in quantum field theory. Fermionic coherent states and Grassmann variables. Ideal Fermi gas and the fermionic Matsubara frequencies. Repulsive fermions: Hartree-Fock approximation. Stoner instability. Attractive fermions: BCS approximation of pairing. Superconductivity. BCS-BEC Crossover. Josephson effect and Josephson junctions.
Path Integral for particles (14 hours)
The quantum propagator and its properties. Feynman path integral construction of the propagator. Propagator of the free particle. Saddle point and stationary phase approximation. Propagator of a particle in a harmonic potential. Wick rotation and thermodynamics. Quantum tunneling.
Path integral for bosons (18 hours)
Partition function of interacting bosons in quantum field theory. Bosonic coherent states. Semiclassical approximation and imaginary time. Coherent states and harmonic potential. Bosonic Matsubara frequencies. Ideal Bose gas of photons. Ideal Bose gas of massive particles. Interacting Bose gas: Bogoliubov spectrum. Dimensional regularization of Gaussian fluctuations. Gross-Pitaevskii equation and superfluid hydrodynamics. Landau criterion and superfluid fraction. Bright and dark solitons. Quantized vortices.
Path integral for fermions (16 hours)
Partition Function of interacting fermions in quantum field theory. Fermionic coherent states and Grassmann variables. Ideal Fermi gas and the fermionic Matsubara frequencies. Repulsive fermions: Hartree-Fock approximation. Stoner instability. Attractive fermions: BCS approximation of pairing. Superconductivity. BCS-BEC Crossover. Josephson effect and Josephson junctions.
- Docente: Silvia Cariani
- Docente: Luca Salasnich