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Quantum gases and condensates

Stephen J. Blundell, Katherine M. Blundell

Published Oct 1, 2009• 0 citations
Publisher

Abstract

Abstract Exchange symmetry affects the occupation of allowed states in quantum gases. If the density of the gas is very low, we can ignore this and forget about exchange symmetry; this is what we do for gases at room temperature. But if the density is high, the effects of exchange symmetry become very important and it really starts to matter whether the particles you are considering are fermions or bosons. This chapter considers quantum gases in detail. It considers non-interacting fermion and boson gases, and discusses Bose-Einstein condensation.

DOI
10.1093/acprof:oso/9780199562091.003.0030

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Abstract Exchange symmetry affects the occupation of allowed states in quantum gases. If the density of the gas is very low, we can ignore this and forget about exchange symmetry; this is what we do for gases at room temperature. But if the density is high, the effects of exchange symmetry become very important and it really starts to matter whether the particles you are considering are fermions or bosons. This chapter considers quantum gases in detail. It considers non-interacting fermion and boson gases, and discusses Bose-Einstein condensation.
Abstract This chapter starts with a discussion of the symmetry conditions which have to be imposed on the wave function of a many-particle system, and the distinction between bosons and fermions. It then examines the consequences of these conditions for simple systems of non-interacting particles (Fermi-Dirac and Bose-Einstein statistics), and in particular, introduces the phenomenon of Bose-Einstein condensation (BEC) in a noninteracting Bose gas. The phenomenon of Cooper pairing of interacting fermions is introduced by a thought-experiment in which one starts from a BEC of tightly bound diatomic molecules made of fermion atoms and gradually weakens the binding. The chapter closes with a description of the various physical systems to be treated in the book, and an account of the phenomenology of superfluidity and superconductivity. An appendix treats the statistical mechanics of a rotating system.
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
At low temperatures the De Broglie wavelength of the gas particles becomes on the order of their average separation, and the effects of their indistinguishability become important. In the absence of a phase transition in the gas, the quantum mechanical Wigner distribution function for a dilute gas of fermions or bosons satisfies the Uehling-Uhlenbeck equation. This equation satisfies an H- theorem with equilibrium solutions being ideal boson or ideal fermion distributions. Navier-Stokes equations can be derived by standard methods. A low temperature gas of weakly interacting bosons undergoes a Bose-Einstein condensation with a macroscopically occupied ground state. A different approach is required to describe the non-equilibrium processes in such a situation. A kinetic equation can be derived for the Bogoliubov excitations in the gas at very low temperatures. The associated hydrodynamic equations are the Landau-Khaltnikov, two fluid equations, and explicit expressions are obtained for the six associated transport coefficients.
This chapter focuses on integrating the distribution function over all the states of conserved Bosons to obtain a relation between particle density and the chemical potential. It describes the phenomenon of Bose-Einstein condensation in a perfect boson gas. The chapter discusses the experimental observation of Bose-Einstein condensation in perfect gases. There are two routes to obtaining Bose-Einstein condensation in a perfect gas. One is to reduce the mass of the particle, thus reducing critical density to a low value at a reasonable temperature. The other route is to lower the temperature to the point where the critical density is so low that the particles are too far apart to interact with each other.
Bose–Einstein condensation is a phenomenon, first proved by Einstein, in which a gas of bosons condenses into the lowest quantum state below a certain critical temperature. It occurs due to the symmetry, under particle exchange, of the boson wave function. The chapter discusses temperature, the Kelvin scale, the concept of absolute zero, and how laser cooling can reach ultralow temperatures. The property of bosons in which they often prefer to be in the same state is seen to apply to lasers, superfluid liquid helium, and to dilute nearly ideal gases. The latter are trapped in magnetic or optical potential wells where the condensation can be observed directly. These large condensate matter waves show quantum interference in a variety of possible experiments.
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.