Lectures

Lecture 1: Failure of classical statistical mechanics

• Mathematica notebook

Additional references

• Cosmic Microwave Background radiation

Lecture 2: Density operator formalism for proper and improper mixed quantum states

• Example: Proper mixtures in spintronics.
• Example: Entangled quantum states, improper mixtures, and decoherence in nanostructures and quantum computers.
• Example: The von Neumann entropy as a measure of state purity.
• Example: Proper mixtures for quantum systems in thermal equilibrium and density operator for microcanonical, canonical, and grand canonical ensembles via correspondence with classical statistical mechanics.
• Example: Density matrix and quantum partition function for a single particle in a box in the quantum canonical ensemble.
• Example: Density matrix and quantum partition function for a linear harmonic oscillator in the quantum canonical ensemble.

Additional references

• Dirac notation and mathematical prerequisites for quantum mechanics (Chapter 1 from L. E. Ballentine: Quantum Mechanics - A modern development, second edition).
• A. Ekert and P. L. Knight, Entangled quantum systems and the Schmidt decomposition, Am. J. Phys. 63, 415 (1995). [PDF]
• Steven Weinberg, Quantum mechanics without state vectors, Phys. Rev. A 90, 042102 (2014). [PDF]

Lecture 3: Master equations for open quantum systems

• Example: Decoherence in quantum Brownian motion.
• Example: Linblad master equation.
• Example: Bloch equation describing spin relaxation and dephasing.

Additional references

• J. K. Gamble and J. F. Lindner, Demystifying decoherence and the master equation of quantum Brownian motion, Am. J. Phys. 77, 244 (2009). [PDF]

Lecture 4: Many-particle wave function and the Hilbert space of identical particles

• Example: Wave functions of 3 fermions and 3 bosons.

Lecture 5: Quantum partition function for many-particle systems in equilibrium

• 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.
• Quantum partition function for non-interacting bosons and fermions in the grand canonical ensemble.
• Example: Equation of state for non-degenerate bosons and fermions.

Additional references

• W. J. Mullin and G. Blaylock, Quantum statistics: Is there an effective fermion repulsion or boson attraction?, Am. J. Phys. 71, 1223 (2003). [PDF]
• G. Cook and R. H. Dickerson, Understanding the chemical potential, Am. J. Phys. 63, 737 (1995). [PDF]

Lecture 6: Degenerate fermions in equilibrium

• Example: Heat capacity of electrons in solids.
• Example: Pauli paramagnetism.
• Example: Landau diamagnetism.
• Example: Stoner ferromagnetism.
• Example: White dwarfs, neutron stars and black holes.

Additional references

• R. Balian and J.-P. Blaizot, Stars and statistical physics: A teaching experience, Am. J. Phys. 67, 1189 (1999). [PDF]
• Stelar evolution: The life and death of stars

Lecture 7: Degenerate bosons in equilibrium

• Example: Bose-Einstein Condensation of free noninteracting bosons.
• Example: Bose-Einstein condensation in ultracold atomic gases.
• Off-diagonal long-range order in the density matrix of Bose-Einstein condensates.
• Example: Heat capacity of phonons in solids.
• Example: Magnons in the Heisenberg model of magnetism.

Additional references

• G. Scharf, On Bose–Einstein condensation, Am. J. Phys. 61, 843 (1993). [PDF]
• W. J. Mullin, The loop-gas approach to Bose–Einstein condensation for trapped particles, Am. J. Phys. 68, 120 (2000). [PDF]
• K. Burnett, M. Edwards, and C. W. Clark, The theory of Bose-Einstein condensation of diluted gases, Phys. Today 52(12), 37 (1999). [PDF]
• E. A. Cornell and C. E. Wieman, Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments, Rev. Mod. Phys. 74, 875 (2002). [PDF]

Lecture 8: Phase transitions in magnetic systems

• Example: Noninteracting spins.
• Thermodynamics of magnetism.
• Abrupt vs. continuous phase transitions in ferromagnet-paramagnet systems and analogy with liquid-gas phase diagram.
• Example: Partition function of the Ising model in one-dimension in an external magnetic field.
• Example: Onsager exact analytical solution vs. Monte Carlo simulations of the Ising model in two-dimensions.

Additional references

• O. Narayan and A. P. Young, Free energies in the presence of electric and magnetic fields, Am. J. Phys. 73, 293 (2005). [PDF]

Lecture 9: Mean-field theories of phase transitions

• Example: Mean-field theory of the Ising model of magnetism.
• Example: Mean-field theory of the XY model of magnetism.
• Landau phenomenological formulation of mean-field theories.
• Landau-Ginzburg mean-field theory and the Ginzburg criterion for the importance of fluctuations.

Additional references

• J. Als‐Nielsen and R. J. Birgeneau, Mean field theory, the Ginzburg criterion, and marginal dimensionality of phase transitions, Am. J. Phys. 45, 554 (1977). [PDF]

Lecture 10: Renormalization group (RG)

• PDF
• Example: RG for 1D and 2D Ising model.
• Example: RG for quantum phase transition in 1D quantum Ising model in transverse magnetic field.
• Example: Numerical RG for 2D Ising model.

Additional references

• K. G. Wilson, Problems in physics with many scales of length, Scientific American 241(2), 140 (1979). [PDF]
• H. J. Maris and L. P. Kadanoff, Teaching the renormalization group, Am. J. Phys. 46, 652 (1978). [PDF]
• M. E. Fisher, Renormalization group theory: Its basis and formulation in statistical physics, Rev. Mod. Phys. 70, 653 (1998). [PDF]
• S. Sachdev and B. Keimer, Quantum criticality, Physics Today 64(2), 29 (2011). [PDF]

Lecture 11: Elements of nonequilibrium statistical physics: Boltzmann equation and Kubo formula

• PDF

Additional references

• N. Borghini, Topics in Nonequilibrium Physics