Abstract This chapter begins by extending the Boltzmann distribution to the case of a system that exchanges particles with its environment. This idea finds direct applications ranging from hemoglobin adsorption to ionization of atoms in stars. But the main applications are to dense “gases” in which the quantum behavior of identical particles comes into play. Examples include conduction electrons in metals and semiconductors; white dwarf and neutron stars; photon gases and thermal radiation from incandescent objects; neutrino and electron-positron production in the early universe; quasiparticles associated with vibrations and magnetic excitations in solids; and Bose-Einstein condensation of ultracold clouds of atoms.
Abstract This chapter begins by extending the Boltzmann distribution to the case of a system that exchanges particles with its environment. This idea finds direct applications ranging from hemoglobin adsorption to ionization of atoms in stars. But the main applications are to dense “gases” in which the quantum behavior of identical particles comes into play. Examples include conduction electrons in metals and semiconductors; white dwarf and neutron stars; photon gases and thermal radiation from incandescent objects; neutrino and electron-positron production in the early universe; quasiparticles associated with vibrations and magnetic excitations in solids; and Bose-Einstein condensation of ultracold clouds of atoms.
Under conditions of high temperature and low density, we shall nd that it does not matter whether classical or quantum mechanical descriptions are used for a system of particles. At high densities and low temperatures this is no longer true, because of the overlap of the particles’ wave functions and quantum mechanics must be used, giving rise to quantum statistics. Under high-density, low-temperature conditions, the properties of a system depend in a crucial way on whether the particles that make up the system are fermions or bosons. Many fascinating phenomena occur in condensed matter as the temperature is lowered. Examples are ferromagnetism, superconductivity, and super uidity. ese important new properties appear fairly abruptly at phase transitions. Progress in the microscopic understanding and description of the behavior of these systems involves quantum mechanics and statistical physics ideas. Applications of quantum statistics are not conned to terrestrial systems and include astrophysical phenomena such as the microwave background radiation from the Big Bang and the mass-radius relationships for white dwarf stars and neutron stars. An important concept in thermal physics is that of entropy, which, as we shall see, increases with time as systems become more disordered. e increase of the entropy of the universe with the passage of time provides what is termed time’s arrow. An interesting and unanswered question arises as to why the entropy of the universe was so low at the beginning of time.
Abstract Quantum statistical mechanics governs metals, semiconductors, and neutron stars. Statistical mechanics spawned Planck’s invention of the quantum, and explains Bose condensation, superfluids, and superconductors. This chapter briefly describes these systems using mixed states, or more formally density matrices, and introducing the properties of bosons and fermions. We discuss in unusual detail how useful descriptions of metals and superfluids can be derived by ignoring the seemingly important interactions between their constituent electrons and atoms. Exercises explore how gregarious bosons lead to superfluids and lasers, how unsociable fermions explain transitions between white dwarfs, neutron stars, and black holes, how one calculates materials properties in semiconductors, insulators, and metals, and how statistical mechanics can explain the collapse of the quantum wavefunction during measurement.
Chapter 15 is devoted to systems of identical particles and the associated quantum statistics. It is shown how the spin (whether integer or half-integer) determines fundamental features of the system’s behavior that cannot be obtained by approximate methods, in particular the exchange effects characteristic of states of two or more identical particles. The spin-symmetry relation is discussed in detail for both the Bose–Einstein and the Fermi–Dirac statistics. Quantum collective phenomena that are a defining feature of condensed matter, including phonons, superconductivity, Bose–Einstein condensates and the quantum Hall effect, are briefly introduced. Since most real quantum systems consist of subsystems in different quantum states that are mixtures of pure states, the chapter includes an introduction to the density matrix, its basic properties and its application to quantum statistics. It concludes with a critical discussion of decoherence and the significance of the collapse of the wave function.
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
Bose–Einstein condensates in dilute atomic gases, which were first realized experimentally in 1995 for rubidium, sodium, and lithium, provide unique opportunities for exploring quantum phenomena on a macroscopic scale. These systems differ from ordinary gases, liquids, and solids in a number of respects, as we shall now illustrate by giving typical values of some physical quantities. The particle density at the centre of a Bose-Einstein condensed atomic cloud is typically 10 13 –10 15 cm −3 . By contrast, the density of molecules in air at room temperature and atmospheric pressure is about 10 19 cm −3 . In liquids and solids the density of atoms is of order 1022 cm −3 , while the density of nucleons in atomic nuclei is about 10 38 cm −3 . To observe quantum phenomena in such low-density systems, the temperature must be of order 10 −5 K or less. This may be contrasted with the temperatures at which quantum phenomena occur in solids and liquids. In solids, quantum effects become strong for electrons in metals below the Fermi temperature, which is typically 10 4 –10 5 K, and for phonons below the De bye temperature, which is typically of order 10 2 K. For the helium liquids, the temperatures required for observing quantum phenomena are of order 1 K. Due to the much higher particle density in atomic nuclei, the corresponding degeneracy temperature is about 10 11 K. The path that led in 1995 to the first realization of Bose–Einstein condensation in dilute gases exploited the powerful methods developed over the past quarter of a century for cooling alkali metal atoms by using lasers.
Bose–Einstein and Fermi–Dirac statistics versus Boltzmann statistics It was remarked in Chapter 2, §2.1, that there are circumstances in which the “Boltzmann statistics” for an ideal gas – the approximation in Eq. (2.6), in which the total partition function is decomposed into a product of single-molecule partition functions, with division by N 1 ! N 2 ! … to correct for particle identity – is inaccurate even if the gas is still ideal (non-interacting particles). That happens when the number of single-particle states within the energy kT of the ground state is no longer very much greater than the number of particles. (Recall Fig. 2.1 and the accompanying discussion.) We noted that when the Boltzmann statistics is no longer an adequate approximation one must distinguish the particles as fermions, no two of which can be in the same single-particle state (counting states of different spin as different), or as bosons, for which there is no restriction on multiple occupancy of the singleparticle states. Fermions are particles of half-odd-integer spin quantum number; they could be composite particles consisting of an odd number of spin-½ particles. Protons, neutrons, and electrons, all of which are of spin ½, are fermions. Bosons are particles of integer spin; they could be spinless, or of spin 1, etc., or could be composites consisting of an even number of fermions. The nuclei of helium atoms of mass number 4 consist of two protons and two neutrons, and the neutral atom has two electrons outside the nucleus; 4 He atoms are bosons.