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Oxford University Press eBooks

Ideal Quantum Gases

Robert H. Swendsen

Published Mar 1, 2012• 0 citations
Publisher

Abstract

This chapter develops the basic equations that will be used to analyse the Fermi-Dirac and Bose-Einstein gases. The representation of many-particle states in terms of products of single-particle states is presented. The reasons for using the quantum grand canonical ensemble are given, and a general expression for the grand canonical partition function is derived. The essential equations for fermions, bosons, and distinguishable particles are developed, and the basic strategy for using them to solve problems is given.

DOI
10.1093/acprof:oso/9780199646944.003.0026

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This chapter develops the basic equations that will be used to analyse the Fermi-Dirac and Bose-Einstein gases. The representation of many-particle states in terms of products of single-particle states is presented. The reasons for using the quantum grand canonical ensemble are given, and a general expression for the grand canonical partition function is derived. The essential equations for fermions, bosons, and distinguishable particles are developed, and the basic strategy for using them to solve problems is given.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
As a preliminary to discussing the properties of Bose–Einstein and Fermi–Dirac gases, this chapter presents the basic quantum equations that underlie both. The process of building N-body states from single-particle states is described, along with notation necessary to simplify the problem. The average number of particles in a given energy state and chemical potential is derived for fermions and bosons. Although all atoms are either bosons or fermions, and therefore indistinguishable, there are nevertheless real systems that are composed of distinguishable particles. Although all atoms are either bosons or fermions, and therefore indistinguishable, there are nevertheless real systems that are composed of distinguishable particles.
We developed a reasonably detailed and comprehensive understanding of boson and fermion statistics from the previous chapter. While this is interesting physics in itself, more important for us is an understand of how the symmetry properties of these particles impact on the entropy associated with them. In this chapter, we start by considering situations where the number of particles in a system is not fixed. Next, the theory developed is used to derive quantum statistics again, but this time without recourse to various approximations. Access to the requisite probability functions enables the Gibbs entropy to be calculated for fermions and bosons .
In this work, improvements are introduced to the current models of the ideal Fermi gas and the ideal Bose gas by incorporating the quantum nature of phase space, which is directly linked to the uncertainty principle. These improved models build upon the recently developed concepts of quantum phase space (QPS) and the QPS representation of quantum mechanics. The Hamiltonian operator for a gas particle and its eigenstates are first determined, and quantum statistical mechanics is used to derive the thermodynamic properties of the ideal gas. Analytic expressions for thermodynamic quantities—including the grand canonical potential, particle number, internal energy, von Neumann entropy, and pressure—are derived, along with the corresponding thermodynamic equations of state for both bosons and fermions. These corrections are particularly significant at low temperatures and in confined volumes, where quantum effects related to system geometry, such as shape and size, become significant. The results also establish a direct link between thermodynamic functions and the quantum statistical variances of momentum. Importantly, the improved models recover well-known classical relations of the ideal gas in the high-temperature and large-volume limits, ensuring consistency with classical physics. By addressing quantum corrections and their thermodynamic implications, this work provides a foundation for further applications in nanoscale systems, quantum gases, low-temperature physics, ultracold physics, and astrophysics.
Abstract This chapter presents the statistical description of quantum gases: bosons and fermions. The coarse-grained distribution function, measuring the average occupation of each quantum level is the fundamental object under study. In thermal equilibrium, gases obey the Bose—Einstein and Fermi—Dirac distributions. Out of equilibrium, a semiclassical description is adopted where the distribution functions evolve as particles and move among levels. For bosons, the presence of other particles in the final level enhances the transitions (stimulated emission), while for fermions they suppress the transition (Pauli exclusion principle). These effects are included semiclassically in the kinetic equation. It is shown that the quantum kinetic equation gives the correct evolution toward the equilibrium distributions. Phonons are worked out in detail, where the phonon–phonon and phonon–electron interactions affect the energy transport. Finally, lasers and the quark—gluon plasma are analysed in the context of the theory.
This chapter investigates a quantum gas in the grand-canonical ensemble with the grand canonical Hamiltonian ensemble. The path integral formalism pioneered by Feynman has proven to be an extremely useful tool for understanding and handling quantum mechanics, quantum field theory, and statistical mechanics. The path integral approach is a powerful tool that considers nonperturbative calculations for the investigation of many-particle systems. It is worth noting that while the path integral approach seems to be quite cumbersome in quantum mechanics, it does provide a powerful tool in quantum field theory. The entire aforementioned derivation is calculated in order to justify the evaluation of the path integral at equal times. The chapter examines the example of an atomic Fermi gas by considering an electron gas. The Fermi liquid concept is the deviation of the electron gas from the ideal gas behavior. It utilizes the so-called diagram technique to calculate some quantities.