Lectures

Lecture 1: Faliure of classical statistical mechanics

• Example: Black-body radiation.

Lecture 2: Mixed states in quantum mechanics and density operator

• Example: Proper mixed states in spintronics.
• Example: Improper mixed states in decoherence of qubits and von Neumann entropy.

Lecture 3: Many-particle wave function and Hilbert space

• Example: Wave function of 3 fermions.
• Example: Wave function of 3 bosons.

Lecture 4: Quantum ensembles and partition function for systems in equilibrium

• Example: Quantum partition function and density matrix for a single particle in a box in a quantum canonical ensemble.
• Example: Quantum partition function and density matrix for a linear harmonic oscillator in a quantum canonical ensemble.
• Example: Quantum partition function for two bosons and two fermions and comparison with classical statistical mechanics.
• Example: "Effective force" between noninteracting bosons and fermions due to Pauli principle.
• Additional reference:
  • W. J. Mullin and G. Blaylock, Quantum statistics: Is there an effective fermion repulsion or boson attraction?, Am. J. Phys. 71, 1223 (2003). [PDF]

Lecture 5: Degenerate bosons in equilibrium

• Example: Heat capacity of phonons in solids.
• Example: Magnons in the Heisenberg model of magnetism.

Lecture 6: Bose-Einstein condensation

• Example: BEC for free noninteracting bosons.
• Example: BEC in ultracold atom gases in traps.

Lecture 7: Degenerate fermions in equilibrium

• Example: Pressure of ideal Fermi vs. ideal Bose gas.
• Example: Heat capacity of electrons in solids.
• Example: Pauli paramagnetism.
• Example: Landau diamagnetism.
• Example: Stoner ferromagnetism.
• Example: Neutron stars.

Lecture 8: Magnetic systems

• Example: Noninteracting spins.
• Thermodynamics of magnetism.
• Example: Partition function of the Ising model in one-dimension.
• Example: Onsager solution and computer simulations of the Ising model in two-dimensions.

Lecture 9: Phase transitions

• Example: Phase diagrams of liquid-gas and paramagnet-ferromagnet systems.
• Thermodynamic equation of state near phase transitions.

Lecture 10: Mean-field theory of phase transitions

• Example: Mean-field theory of the Ising model of magnetism.
• Example: Mean-field theory of the Heisenberg model of magnetism and upper critical dimensionality.
• Example: Landau theory and the origin of its failure for two-dimensional Ising model.

Lecture 11: Critical phenomena and renormalization group (RG)

• Universality and scaling relations
• Example: Percolation as geometrical phase transition.
• Example: RG for percolation.
• Example: RG for 1D Ising model.
• Example: Niemeijer-van Leeuwen RG in real space for 2D Ising model.
• Additional references:
  • Fisher ...