2016 Quantum Statistical Mechanics

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Academic unit or major
Graduate major in Materials Science and Engineering
Kajihara Masanori  Nakatsuji Kan  Gohda Yoshihiro 
Class Format
Media-enhanced courses
Day/Period(Room No.)
Tue1-2(J234)  Fri1-2(J234)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
Access Index

Course description and aims

Quantum mechanics is reviewed as an introduction to statistical mechanics. On the basis of the Boltzmann's principle and Lagrange multipliers, Boltzmann and Gibbs factors are derived.
Using basic principles of statistical mechanics, entropies due to lattice vibrations, electrons, spins are evaluated. Heat capacities by Einstein and Debye models are discussed. The Fermi-Dirac statistics and Bose-Einstein statistics are introduced, where the Pauli's exclusion principle are discussed.

Student learning outcomes

To understand roles of quantum mechanics in thermal physics and statistical mechanics.
To understand the differences among microcanonical, canonical and grand canonical ensembles.
To understand the differences in the partition function between the particle-distinguishable and particle-indistinguishable cases.
To understand that spins govern the statistics of quantum particles.
To understand characteristics of Fermions and Bosons by the Pauli's exclusion principle.


Boltzmann's principle, partition functions, creation/annihilation operators, Debye model, Fermions, Bosons

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

The lectures are complemented with exercises.

Course schedule/Required learning

  Course schedule Required learning
Class 1 State vectors and operators Understand the Hilbert space and the expectation values of operators as physical observables.
Class 2 Canonical commutation relations Understand the uncertainty principle through commutation relations.
Class 3 Angular momenta and spins Understand the theory of angular momenta.
Class 4 Boltzmann's principle and Lagrange multipliers Understand how to derive the partition functions using the method of Lagrange multiplier based on the Boltzmann's principle.
Class 5 Partition functions and thermodynamic functions Students must be able to derive internal energy, Helmholtz energy and entropy.
Class 6 Fluctuations and response functions Understand the Legendre transformation and relation between fluctuations and response functions.
Class 7 Creation and annihilation operators Understand creation and annihilation operators in quantum mechanics.
Class 8 Specific heat -Einstein model- Students must be able to derive specific heat of solids using Einstein model from the knowledge of quantum harmonic oscillators.
Class 9 Specific heat -Debye model- Understand how to derive specific heat of solids using Debye model and its difference from Einstein model.
Class 10 Ideal gases Students must be able to derive the partition function of the perfect gas.
Class 11 Fermions Explain Fermi-Dirac statistics and the free-electron gas.
Class 12 Bosons Explain Bose-Einstein statistics and the Bose-Einstein condensation.
Class 13 Magnetization in localized spin systems Students must be able to derive specific heats, magnetization, etc. from the partition function for the localized spin system.
Class 14 Mean-field approximation and phase transitions Understand the mean-field approximation in the Heisenberg model and Landau theory of phase transitions.
Class 15 Phase equilibria and solution models Understand the law of mass action, the Clausius-Clapeyron relation, the entropy of configuration and the solubility gap.


Lecture notes are distributed.

Reference books, course materials, etc.

Sakurai: Modern Quantum Mechanics,
Kittel: Thermal Physics,
Rushbrooke: Introduction to Statistical Mechanics

Assessment criteria and methods

Learning achievement is evaluated by examinations.

Related courses

  • MAT.M407 : Advanced Solid State Physics
  • MAT.M409 : Themodynamics for Phase Equilibria
  • MAT.M411 : Phase Transformation and Microstructure Control
  • MAT.C406 : Advanced Course of Magnetism

Prerequisites (i.e., required knowledge, skills, courses, etc.)


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