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Oxford University Press eBooks

Quantum Canonical Ensemble

Robert H. Swendsen

Published Mar 1, 2012• 0 citations
Publisher

Abstract

This chapter derives the canonical ensemble, which is arguably the most important concept in quantum statistical mechanics. The associated partition function leads to a quantum statistical mechanics expression for entropy. The origin of the Nernst postulate (third law of thermodynamics) then follows directly. The important special cases of two-level systems and simple harmonic oscillators are discussed in detail.

DOI
10.1093/acprof:oso/9780199646944.003.0023

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This chapter derives the canonical ensemble, which is arguably the most important concept in quantum statistical mechanics. The associated partition function leads to a quantum statistical mechanics expression for entropy. The origin of the Nernst postulate (third law of thermodynamics) then follows directly. The important special cases of two-level systems and simple harmonic oscillators are discussed in detail.
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.
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.
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 .
The author presents his experience in teaching at a graduate level the quantal exposition of a new non-statistically based paradigm of physics and thermodynamics. This paradigm, called the Unified Quantum Theory of Mechanics and Thermodynamics, applies to all systems large or small (including one particle systems) either in a state of thermodynamic (i.e. stable) equilibrium or not in a state of thermodynamic equilibrium. It uses as its primitives inertial mass, force, and time and introduces the laws of thermodynamics in the most unambiguous and general formulations found in the literature. Starting with a precise definition of system and of state followed by statements and corollaries of the laws of thermodynamics, the thermodynamic formalism is developed without circularity and ambiguity. In this quantal exposition of the new paradigm, a brief review of the formalism of thermodynamics as a general science not limited to stable equilibrium and large (macroscopic) systems as well as a very brief summary of the three prevalent formalisms in classical physics are presented followed by a presentation and development of solutions for a number of elementary problems in quantum physics (e.g., a particle in a box, a harmonic oscillator, a rigid rotor, etc.). These solutions and the maximum entropy principle are then used in a constrained optimization to develop the canonical and grand canonical distributions for Fermi-Dirac and Bose-Einstein types of particles, i.e. for fermions and bosons. This is done without the use of analogies between statistical and thermodynamic results and without additional hypotheses such as the ergodic hypothesis of statistical mechanics. These distributions are then employed under various assumptions (i.e. the Boltzmann, constant-potential, point-particle, and continuous eigenvalue-spectrum approximations) to derive the corresponding thermodynamic property expressions for perfect, semi-perfect (ideal), and Sommerfeld gases as well as for mixtures of ionized and dissociated gases. In a similar fashion but with a change from a single- to a multi-particle partition function and with the addition of various inter-particle potentials for two-particle interactions (e.g., the Lennard-Jones potential, the square-well potential, etc.), expressions for the thermodynamic properties of dense gases are developed and presented.An initial version of this paper was published inJuly of 2006 in the proceedings of ECOS’06, AghiaPelagia, Crete, Greece.
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.