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== Lecture 1: Faliure of classical statistical mechanics: Black-body radiation ==
== Lecture 1: Failure of classical statistical mechanics on black-body radiation problem ==
*[[Media:black_body_bkn.nb|Mathematica notebook]]


== Lecture 2: Mixed states in quantum mechanics and density operator ==
===Additional references===
*[http://star-www.st-and.ac.uk/~kdh1/ce/ce04.pdf Matter-radiation decoupling and the cosmic microwave background]
*C. Andersen, C. A. Rosenstroem, and O. Ruchayskiy, ''How bright was the Big Bang?'', Am. J. Phys. '''87''', 395 (2019). [https://doi.org/10.1119/1.5092705 [PDF]]


== Lecture 3: Many-particle wave functions and Hilbert space ==
== Lecture 2: Density operator formalism for proper and improper mixed quantum states ==
*[[Media:PHYS813_lecture2_density_operator_formalism.pdf|PDF]]
* 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 canonical ensemble.
* Example: Density matrix and quantum partition function for a linear harmonic oscillator in canonical ensemble.


== Lecture 4: Ensembles and partition function for systems in equilibrium ==
===Postulates of Quantum Mechanics in a Nutshell===
* Example: Partition function for two bosons and two fermions.
<pre>
1. Where things happen: HILBERT SPACE
 
2. Combining Quantum Systems: TENSOR PRODUCT OF VECTORS, MATRICES AND OF HILBERT SPACES
 
3. Time Evolution (Dynamics): SCHRÖDINGER EQUATION
 
4. Information extraction: MEASUREMENTS
</pre>
 
=== Additional references ===
* [https://catalogue.library.cern/api/files/1d9e2170-5f3f-43de-8b6a-d8d59b791e09/2.%20Back%20matter.pdf?download Overview of linear algebra, using Dirac notation, for quantum (mechanics, computing and statistical mechanics) courses]
*[https://www.worldscientific.com/doi/suppl/10.1142/9038/suppl_file/9038_chap01.pdf Mathematical Prerequsites] from L. E. Ballentine, ''Quantum Mechanics: A Modern Development'' (World Scientific, Singapore, 2014).
*S. Weinberg, ''Quantum mechanics without state vectors'', Phys. Rev. A '''90''', 042102 (2014). [http://dx.doi.org/10.1103/PhysRevA.90.042102 [PDF]]
*A. Ekert and P. L. Knight, ''Entangled quantum systems and the Schmidt decomposition'', Am. J. Phys. '''63''', 415 (1995). [http://dx.doi.org/10.1119/1.17904 [PDF]]
*J. Kincaida, K. McLelland, and M. Zwolak, ''Measurement-induced decoherence and information in double-slit interference'', Am. J. Phys. '''84''', 522 (2016). [http://dx.doi.org/10.1119/1.4943585 [PDF]]
*W. H. Zurek, ''Decoherence and the transition from quantum to classical revisited'', arXiv:quant-ph/0306072 (2003). [https://arxiv.org/abs/quant-ph/0306072 [PDF]]
 
== Lecture 3: Many-particle wave function and the Hilbert space of identical particles ==
*[[Media:PHYS813_lecture3_identical_particles.pdf|PDF]]
* Example: Wave functions of 3 fermions and 3 bosons.
* Example: Ortho vs. para hydrogen H<sub>2</sub> molecules.
 
===Additional references===
*C. F. Roos, A. Alberti, D. Meschede, P. Hauke, and H. Häffner, ''Revealing quantum statistics with a pair of distant atoms'', Phys. Rev. Lett. '''119''', 160401 (2017). [https://doi.org/10.1103/PhysRevLett.119.160401 [PDF]]
*N. Vaytet, K. Tomida, and G. Chabrier, ''On the role of the H<sub>2</sub> ortho:para ratio in gravitational collapse during star formation'', Astronomy & Astrophysics '''563''', A85 (2014). [https://www.aanda.org/articles/aa/pdf/2014/03/aa22855-13.pdf [PDF]]
 
== Lecture 4: Quantum partition function for noninteracting many-particle systems==
*[[Media:PHYS813_lecture4_partition_function_for_many_particles.pdf|PDF]]
* 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.
* Example: "Effective force" between noninteracting bosons and fermions due to Pauli principle.
* Example: Quantum partition function for noninteracting bosons and fermions in the grand canonical ensemble.
* Example: Equation of state for non-degenerate bosons and fermions.


== Lecture 5: Noninteracting bosons in equilibrium ==
=== Additional references ===  
* Example: Phonons in solids.
* W. J. Mullin and G. Blaylock, ''Quantum statistics: Is there an effective fermion repulsion or boson attraction?'', Am. J. Phys. '''71''', 1223 (2003). [http://dx.doi.org/10.1119/1.1590658 [PDF]]
 
* G. Cook and R. H. Dickerson, ''Understanding the chemical potential'', Am. J. Phys. '''63''', 737 (1995). [http://dx.doi.org/10.1119/1.17844 [PDF]]
== Lecture 6: Bose-Einstein condensation ==
*Example: BEC in noninteracting gases.
*Example: BEC in atomic traps.


== Lecture 7: Noninteracting fermions in equilibrium ==
== Lecture 5: Degenerate fermions ==
*Example: Pressure of ideal Fermi vs. ideal Bose gas.
*[[Media:PHYS813_lecture5_degenerate_fermions.pdf|PDF]]
*Example: Heat capacity of electrons in solids.
*Example: Heat capacity of electrons in solids.
*Example: Pauli paramagnetism.
*Example: Pauli paramagnetism.
*Example: Landau diamagnetism.
*Example: Landau diamagnetism.
*Example: Neutron stars.
*Example: Stoner ferromagnetism.
*Example: [[Media:stars_phys813.pdf|White dwarfs, neutron and quark stars and black holes]].
 
=== Additional references ===
*[https://map.gsfc.nasa.gov/universe/rel_stars.html Stelar evolution: The life and death of stars]
*[https://ras.ac.uk/news-and-press/news/swallowed-torn-or-live-how-earth-will-fare-when-sun-dies How Earth will fare when the Sun dies]
* R. Balian and J.-P. Blaizot, ''Stars and statistical physics: A teaching experience'', Am. J. Phys. '''67''', 1189 (1999). [http://dx.doi.org/10.1119/1.19105 [PDF]]
*R. Fantoni, ''White-dwarf equation of state and structure: the effect of temperature'', J. Stat. Mech.  113101 (2017). [https://iopscience.iop.org/article/10.1088/1742-5468/aa9339 [PDF]]
*[[Media:fermi_letter_to_dirac.jpg|Fermi writes to Dirac about "their" distribution function]] and his own [https://arxiv.org/pdf/cond-mat/9912229.pdf paper] from 1926.
 
== Lecture 6: Degenerate bosons ==
* [[Media:PHYS813_lecture6_degenerate_bosons.pdf|PDF]]
* Example: Bose-Einstein Condensation of free noninteracting bosons.
* Example: [[Media:bec_phys813.pdf|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.
 
===Additional references===
*G. Scharf, ''On Bose–Einstein condensation'', Am. J. Phys. '''61''', 843 (1993). [http://dx.doi.org/10.1119/1.17416 [PDF]]
*W. J. Mullin, ''The loop-gas approach to Bose–Einstein condensation for trapped particles'', Am. J. Phys. '''68''', 120 (2000). [http://ajp.aapt.org.proxy.nss.udel.edu/resource/1/ajpias/v68/i2/p120_s1[PDF]]
*K. Burnett, M. Edwards, and C. W. Clark, ''The theory of Bose-Einstein condensation of diluted gases'', Phys. Today '''52(12)''', 37 (1999). [http://bec.nist.gov/PDF/theoryBEC.pdf [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). [http://rmp.aps.org/abstract/RMP/v74/i3/p875_1[PDF]]
 
== Lecture 7: Phase transitions ==
*[[Media:PHYS813_lecture7_phase_transitions.pdf|PDF]]
* Thermodynamics of magnetism.
* Exchange interaction.
* Quantum and classical Heisenberg or Ising ferro- and anti-ferromagnetic models.
* Example: Abrupt vs. continuous phase transitions in ferromagnet-paramagnet systems and analogy with liquid-gas phase diagram.
* Example: Phase transitions in quantum fluids, quantum chromodynamics and cosmology.
* Example: 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). [http://dx.doi.org/10.1119/1.1819934 [PDF]]
* A. Mazumdar and G. White, ''Review of cosmic phase transitions: their significance and experimental signatures'', Rep. Prog. Phys. '''82''', 076901 (2019). [https://iopscience.iop.org/article/10.1088/1361-6633/ab1f55 [PDF]]
* D. Vollhardt and P. Wölfle, ''Superfluid Helium 3: Link between condensed matter physics and particle physics'', [https://arxiv.org/pdf/cond-mat/0012052.pdf arXiv:cond-mat/0012052].
* M. J. Aschwanden ''et al.'', ''25 years of self-organized criticality: Solar and astrophysics'', Space Sci. Rev. '''198''', 47  (2016). [https://link.springer.com/content/pdf/10.1007/s11214-014-0054-6.pdf [PDF]]
* A. Scheie, P. Laurell, A. M. Samarakoon, B. Lake, S. E. Nagler, G. E. Granroth, S. Okamoto, G. Alvarez, and D. A. Tennant, ''Witnessing entanglement in quantum magnets using neutron scattering'', Phys. Rev. B '''103''', 224434 (2021). [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.224434 [PDF]]
 
== Lecture 8: Mean-field theory of phase transitions ==
*[[Media:PHYS813_lecture8_mean_field_theory.pdf|PDF]]
* Example: Mean-field theory of the Ising model of magnetism.
* Example: Mean-field theory of the XY model of magnetism.
* Example: Mean-field theory of the Hubbard model.
* Landau phenomenological formulation of mean-field theories.
* Landau-Ginzburg mean-field theory and the Ginzburg criterion for the importance of fluctuations.


== Lecture 8: Magnetic systems ==
===Additional references===
* M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov, ''Magnetic 2D materials and heterostructures'', Nat. Nanotech. '''14''', 408 (2019). [https://www.nature.com/articles/s41565-019-0438-6 [PDF]]
* T. Olsen, ''Theory and simulations of critical temperatures in CrI3 and other 2D materials: easy-axis magnetic order and easy-plane Kosterlitz-Thouless transitions'', MRS Commun. '''9''', 1142 (2019). [https://link.springer.com/article/10.1557%2Fmrc.2019.117 [PDF]]
*A. Bedoya-Pinto ''et al.'', ''Intrinsic 2D-XY ferromagnetism in a van der Waals monolayer'', Science '''374''', 616 (2021). [https://www.science.org/doi/10.1126/science.abd5146 [PDF]]
*Y. Claveau, B. Arnaud, and S. Di Matteo, ''Mean-field solution of the Hubbard model: the magnetic phase diagram'', Eur. J. Phys. '''35''', 035023 (2014). [https://iopscience.iop.org/article/10.1088/0143-0807/35/3/035023/meta [PDF]]


== Lecture 9: Phase transitions in the Ising model of magnetism ==
== Lecture 9: Renormalization group (RG) theory of phase transitions==
* [[Media:rg_phys813.pdf|Tutorial on renormalization group applied to classical and quantum critical phenomena]]
* 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.


== Lecture 10: Mean-field theory of phase transitions ==
===Additional references===
* K. G. Wilson, ''Problems in physics with many scales of length'', Scientific American '''241(2)''', 140 (1979). [[Media:WILSON=problems_in_physics_with_many_scales_of_length.pdf|[PDF]]]
* H. J. Maris and L. P. Kadanoff, ''Teaching the renormalization group'', Am. J. Phys. '''46''', 652 (1978). [http://dx.doi.org/10.1119/1.11224 [PDF]]
* M. E. Fisher, ''Renormalization group theory: Its basis and formulation in statistical physics'', Rev. Mod. Phys. '''70''', 653 (1998). [http://link.aps.org/doi/10.1103/RevModPhys.70.653 [PDF]]
* S. Sachdev and B. Keimer, ''Quantum criticality'', Physics Today '''64(2)''', 29 (2011). [http://dx.doi.org/10.1063/1.3554314 [PDF]]
*[https://www.quantamagazine.org/how-renormalization-saved-particle-physics-20200917/ How mathematical ‘hocus-pocus’ saved particle physics] (popular and historical account of renormalization as perhaps the single most important advance  in theoretical physics in 50 years).


== Lecture 11: Renormalization group ==
== Lecture 10: Elements of nonequilibrium statistical physics: Boltzmann equation and Kubo formula ==
* [[Media:boltzmann_kubo.pdf|Elements of nonequilibrium statistical physics: Boltzmann equation and Kubo formula]]
===Additional references===
* L. Szunyogh, Kubo formula for electronic transport [[Media:SZUNYOGH=theory_of_electronic_transport.pdf|[PDF]]]


== Lecture 12: Boltzmann (semiclassical) theory of linear response ==
== Lecture 11: Elements of nonequilibrium statistical physics: Master equations for density operator of open quantum system ==
* Example: Bloch equation describing thermal relaxation and decoherence of spin-1/2.
* Example: Decoherence in quantum Brownian motion.


== Lecture 13: Kubo (quantum) theory of linear response ==
=== 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). [http://dx.doi.org/10.1119/1.3043847 [PDF]]
* D. A. Lidar, ''Lecture notes on the theory of open quantum systems'', [https://arxiv.org/abs/1902.00967 arXiv:1902.00967].

Latest revision as of 12:22, 18 February 2025

Lecture 1: Failure of classical statistical mechanics on black-body radiation problem

Additional references

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

  • PDF
  • 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 canonical ensemble.
  • Example: Density matrix and quantum partition function for a linear harmonic oscillator in canonical ensemble.

Postulates of Quantum Mechanics in a Nutshell

1. Where things happen: HILBERT SPACE

2. Combining Quantum Systems: TENSOR PRODUCT OF VECTORS, MATRICES AND OF HILBERT SPACES

3. Time Evolution (Dynamics): SCHRÖDINGER EQUATION

4. Information extraction: MEASUREMENTS

Additional references

Lecture 3: Many-particle wave function and the Hilbert space of identical particles

  • PDF
  • Example: Wave functions of 3 fermions and 3 bosons.
  • Example: Ortho vs. para hydrogen H2 molecules.

Additional references

  • C. F. Roos, A. Alberti, D. Meschede, P. Hauke, and H. Häffner, Revealing quantum statistics with a pair of distant atoms, Phys. Rev. Lett. 119, 160401 (2017). [PDF]
  • N. Vaytet, K. Tomida, and G. Chabrier, On the role of the H2 ortho:para ratio in gravitational collapse during star formation, Astronomy & Astrophysics 563, A85 (2014). [PDF]

Lecture 4: Quantum partition function for noninteracting many-particle systems

  • PDF
  • 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.
  • Example: Quantum partition function for noninteracting 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 5: Degenerate fermions

Additional references

Lecture 6: Degenerate bosons

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 7: Phase transitions

  • PDF
  • Thermodynamics of magnetism.
  • Exchange interaction.
  • Quantum and classical Heisenberg or Ising ferro- and anti-ferromagnetic models.
  • Example: Abrupt vs. continuous phase transitions in ferromagnet-paramagnet systems and analogy with liquid-gas phase diagram.
  • Example: Phase transitions in quantum fluids, quantum chromodynamics and cosmology.
  • Example: 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]
  • A. Mazumdar and G. White, Review of cosmic phase transitions: their significance and experimental signatures, Rep. Prog. Phys. 82, 076901 (2019). [PDF]
  • D. Vollhardt and P. Wölfle, Superfluid Helium 3: Link between condensed matter physics and particle physics, arXiv:cond-mat/0012052.
  • M. J. Aschwanden et al., 25 years of self-organized criticality: Solar and astrophysics, Space Sci. Rev. 198, 47 (2016). [PDF]
  • A. Scheie, P. Laurell, A. M. Samarakoon, B. Lake, S. E. Nagler, G. E. Granroth, S. Okamoto, G. Alvarez, and D. A. Tennant, Witnessing entanglement in quantum magnets using neutron scattering, Phys. Rev. B 103, 224434 (2021). [PDF]

Lecture 8: Mean-field theory of phase transitions

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

Additional references

  • M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov, Magnetic 2D materials and heterostructures, Nat. Nanotech. 14, 408 (2019). [PDF]
  • T. Olsen, Theory and simulations of critical temperatures in CrI3 and other 2D materials: easy-axis magnetic order and easy-plane Kosterlitz-Thouless transitions, MRS Commun. 9, 1142 (2019). [PDF]
  • A. Bedoya-Pinto et al., Intrinsic 2D-XY ferromagnetism in a van der Waals monolayer, Science 374, 616 (2021). [PDF]
  • Y. Claveau, B. Arnaud, and S. Di Matteo, Mean-field solution of the Hubbard model: the magnetic phase diagram, Eur. J. Phys. 35, 035023 (2014). [PDF]

Lecture 9: Renormalization group (RG) theory of phase transitions

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]
  • How mathematical ‘hocus-pocus’ saved particle physics (popular and historical account of renormalization as perhaps the single most important advance in theoretical physics in 50 years).

Lecture 10: Elements of nonequilibrium statistical physics: Boltzmann equation and Kubo formula

Additional references

  • L. Szunyogh, Kubo formula for electronic transport [PDF]

Lecture 11: Elements of nonequilibrium statistical physics: Master equations for density operator of open quantum system

  • Example: Bloch equation describing thermal relaxation and decoherence of spin-1/2.
  • Example: Decoherence in quantum Brownian motion.

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]
  • D. A. Lidar, Lecture notes on the theory of open quantum systems, arXiv:1902.00967.
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