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Foundations of Statistical Mechanics
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Module 6
Quantum Statistical Mechanics Formalism
1
The Density Matrix and Von Neumann Equation
Define the quantum density matrix (operator) and derive its equation of motion, the von Neumann equation.
2
Density Matrices and Statistical Ensembles
Construct the density matrices for pure and mixed states, and for the microcanonical, canonical, and grand canonical ensembles.
3
Quantum Partition Functions: Tracing the Statistical Operator
Express quantum partition functions as traces of the corresponding statistical operator (e.g., Z = Tr(e^{-βH})).
Deriving Fermi-Dirac and Bose-Einstein Distributions
Derive the Fermi-Dirac and Bose-Einstein distribution functions for average occupation numbers in the grand canonical ensemble.
5
Canonical Partition Function for Quantum Harmonic Oscillators
Compute the canonical partition function for a system of non-interacting quantum harmonic oscillators.
6
Planck's Law from Bosonic Photon Gas
Derive the Planck distribution for blackbody radiation by treating photons as a gas of non-conserved bosons.
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Thermodynamic Equivalence and Ensemble Choice
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Ideal Fermi Gases