
Contents
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24.1 Derivation of the QM Canonical Ensemble 24.1 Derivation of the QM Canonical Ensemble
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24.2 Thermal Averages and the Average Energy 24.2 Thermal Averages and the Average Energy
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24.3 The Quantum Mechanical Partition Function 24.3 The Quantum Mechanical Partition Function
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24.4 The Quantum Mechanical Entropy 24.4 The Quantum Mechanical Entropy
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24.5 The Origin of the Third Law of Thermodynamics 24.5 The Origin of the Third Law of Thermodynamics
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24.6 Derivatives of Thermal Averages 24.6 Derivatives of Thermal Averages
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24.7 Factorization of the Partition Function 24.7 Factorization of the Partition Function
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24.8 Special Systems 24.8 Special Systems
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24.9 Two-Level Systems 24.9 Two-Level Systems
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24.9.1 Energies 0 and ϵ 24.9.1 Energies 0 and ϵ
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24.9.2 Spin-One-Half 24.9.2 Spin-One-Half
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24.10 Simple Harmonic Oscillator 24.10 Simple Harmonic Oscillator
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24.11 Einstein Model of a Crystal 24.11 Einstein Model of a Crystal
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24.12 Problems 24.12 Problems
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PROBLEM 24.1 Quantum statistical mechanics: A spin in a magnetic field PROBLEM 24.1 Quantum statistical mechanics: A spin in a magnetic field
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PROBLEM 24.2 A computer simulation of a two-level system Quantum statistical mechanics: A spin in a magnetic field PROBLEM 24.2 A computer simulation of a two-level system Quantum statistical mechanics: A spin in a magnetic field
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PROBLEM 24.3 Quantum statistical mechanics: Another two-level system PROBLEM 24.3 Quantum statistical mechanics: Another two-level system
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PROBLEM 24.4 Derivation of SC for N two-level systems PROBLEM 24.4 Derivation of SC for N two-level systems
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PROBLEM 24.5 Quantum statistical mechanics: Return of the simple harmonic oscillator PROBLEM 24.5 Quantum statistical mechanics: Return of the simple harmonic oscillator
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PROBLEM 24.6 A Monte Carlo computer simulation of a quantum SHO PROBLEM 24.6 A Monte Carlo computer simulation of a quantum SHO
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PROBLEM 24.7 Quantum statistical mechanics: A many-spin system PROBLEM 24.7 Quantum statistical mechanics: A many-spin system
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PROBLEM 24.8 Quantum statistical mechanics: A many-spin system (This is a slightly more difficult version of the previous problem). PROBLEM 24.8 Quantum statistical mechanics: A many-spin system (This is a slightly more difficult version of the previous problem).
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PROBLEM 24.9 A diatomic ideal gas Quantum statistical mechanics—but only when needed PROBLEM 24.9 A diatomic ideal gas Quantum statistical mechanics—but only when needed
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PROBLEM 24.10 Another diatomic ideal gas—entirely classical this time, but in two dimensions PROBLEM 24.10 Another diatomic ideal gas—entirely classical this time, but in two dimensions
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PROBLEM 24.11 Two-level quantum systems PROBLEM 24.11 Two-level quantum systems
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24 Quantum Canonical Ensemble
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Published:December 2019
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Abstract
This chapter introduces the quantum mechanical canonical ensemble, which is used for the majority of problems in quantum statistical mechanics. The ensemble is derived and analogies with the classical ensemble are presented. A useful expression for the quantum entropy is derived. The origin of the Third Law is explained. The relationship between fluctuations and derivatives found in classical statistical mechanics is shown to have counterparts in quantum statistical mechanics. The factorization of the partition function is re-introduced as the best trick in quantum statistical mechanics. Due to their importance in later chapters, basic calculations of the properties of two-level systems and simple harmonic oscillators are derived.
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