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.
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
✔ Specialist skills | ✔ Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
The lectures are complemented with exercises.
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. |
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Lecture notes are distributed.
Sakurai: Modern Quantum Mechanics,
Kittel: Thermal Physics,
Rushbrooke: Introduction to Statistical Mechanics
Learning achievement is evaluated by examinations.
None.