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.
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.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
Thermodynamics is used to describe the bulk properties of matter in or near equilibrium. Many scientists, notably Boyle, Carnot, Clausius, Gay-Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the development of the field over three centuries. Quantities such as mass, pressure, energy, and so on are readily defined and measured. Classical statistical mechanics attempts to understand thermodynamics by the application of classical mechanics to the microscopic particles making up the system. Great progress in this field was made by physicists like Boltzmann and Maxwell. Temperature, entropy, particle number, and chemical potential are thus understandable in terms of the microscopic nature of matter. Classical mechanics is inadequate in many circumstances however, and ultimately must be replaced by quantum mechanics. In fact, the ultraviolet catastrophe encountered by the application of classical mechanics and electromagnetism to blackbody radiation was one of the problems that led to the development of quantum theory. The development of quantum statistical mechanics was achieved by a number of twentieth century physicists, most notably Planck, Einstein, Fermi, and Bose. The purpose of this chapter is to give a mini-review of the basic concepts of quantum statistical mechanics as applied to noninteracting systems of particles. This will set the stage for the functional integral representation of the partition function, which is a cornerstone of modern relativistic quantum field theory and the quantum statistical mechanics of interacting particles and fields. Ensembles One normally encounters three types of ensemble in equilibrium statistical mechanics. The microcanonical ensemble is used to describe an isolated system that has a fixed energy E , a fixed particle number N , and a fixed volume V .
Abstract 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.
Review of quantum statistical mechanicsThermodynamics is used to describe the bulk properties of matter in or near equilibrium.Many scientists, notably Boyle, Carnot, Clausius, Gay-Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the development of the field over three centuries.Quantities such as mass, pressure, energy, and so on are readily defined and measured.Classical statistical mechanics attempts to understand thermodynamics by the application of classical mechanics to the microscopic particles making up the system.Great progress in this field was made by physicists like Boltzmann and Maxwell.Temperature, entropy, particle number, and chemical potential are thus understandable in terms of the microscopic nature of matter.Classical mechanics is inadequate in many circumstances however, and ultimately must be replaced by quantum mechanics.In fact, the ultraviolet catastrophe encountered by the application of classical mechanics and electromagnetism to blackbody radiation was one of the problems that led to the development of quantum theory.The development of quantum statistical mechanics was achieved by a number of twentieth century physicists, most notably Planck, Einstein, Fermi, and Bose.The purpose of this chapter is to give a mini-review of the basic concepts of quantum statistical mechanics as applied to noninteracting systems of particles.This will set the stage for the functional integral representation of the partition function, which is a cornerstone of modern relativistic quantum field theory and the quantum statistical mechanics of interacting particles and fields. EnsemblesOne normally encounters three types of ensemble in equilibrium statistical mechanics.The microcanonical ensemble is used to describe an isolated system that has a fixed energy E, a fixed particle number N , and a fixed 1
An energy level consists of all microscopic states of a system with the same energy. The energy levels of a system depend on the number of particles and the volume. At each instance in time, many different quantum states are represented in the various systems in the ensemble. A microcanonical ensemble describes an isolated system which has a fixed energy, volume and number of particles. The chapter describes certain relationships between fluctuations in thermodynamical variables and some other thermodynamical entities. One case of particular interest is when each subsystem is composed of one single particle, for instance very weakly interacting particles adsorbed on a surface or the particles in an ideal gas. The chapter presents more strict derivations of the probability expressions in the canonical, isobaric-isothermal and grand canonical ensembles. Volume and number of particles can also be considered as being mechanical variables. Entropy and temperature are examples of nonmechanical variables.