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Quantum Statistical Field TheoryAn Introduction to Schwinger's Variational Method with Green's Function Nanoapplications, Graphene and Superconductivity$
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Norman J. Morgenstern Horing

Print publication date: 2017

Print ISBN-13: 9780198791942

Published to Oxford Scholarship Online: January 2018

DOI: 10.1093/oso/9780198791942.001.0001

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Quantum Mechanical Ensemble Averages and Statistical Thermodynamics

Quantum Mechanical Ensemble Averages and Statistical Thermodynamics

Chapter:
(p.150) 6 Quantum Mechanical Ensemble Averages and Statistical Thermodynamics
Source:
Quantum Statistical Field Theory
Author(s):

Norman J. Morgenstern Horing

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198791942.003.0006

Chapter 6 introduces quantum-mechanical ensemble theory by proving the asymptotic equivalence of the quantum-mechanical, microcanonical ensemble average with the quantum grand canonical ensemble average for many-particle systems, based on the method of Darwin and Fowler. The procedures involved identify the grand partition function, entropy and other statistical thermodynamic variables, including the grand potential, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy and their relations in accordance with the fundamental laws of thermodynamics. Accompanying saddle-point integrations define temperature (inverse thermal energy) and chemical potential (Fermi energy). The concomitant emergence of quantum statistical mechanics and Bose–Einstein and Fermi–Dirac distribution functions are discussed in detail (including Bose condensation). The magnetic moment is derived from the Helmholtz free energy and is expressed in terms of a one-particle retarded Green’s function with an imaginary time argument related to inverse thermal energy. This is employed in a discussion of diamagnetism and the de Haas-van Alphen effect.

Keywords:   quantum statistical mechanics, Bose–Einstein, Fermi–Dirac, Bose condensation, quantum-mechanical ensemble theory, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy, magnetic moment, temperature, chemical potential, Fermi energy, microcanonical ensemble, grand canonical ensemble, grand partition function, grand potential, entropy, diamagnetism, magnetic moment, de Haas-van Alphen effect

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