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Cambridge University Press eBooks

Review of quantum statistical mechanics

Joseph I. Kapusta, Charles Gale

Published Jul 13, 2023• 1 citation
Publisher

Abstract

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

DOI
10.1017/9781009401968.002

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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
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
We establish a rather deep relation between statistical mechanics and quantum mechanics at an imaginary time. For the quantum field theory, this implies a relation between finite temperature quantum field theory in Minkowski space and quantum field theory in Euclidean space with one compact Euclidean time direction. The quantum mechanical language being a proper formulation of statistical physics as a molecular theory is basically the study of quantum equilibrium as well as non-equilibrium systems. A quantum system found in a thermodynamic equilibrium state with an external classical system (thermostat or thermal reservoir) may be characterized by fixed values of thermodynamic parameters of the system, say, for example, temperature T , volume V and number of particles N . In this way the total energy of the thermostat is assumed to be much greater than the energy of the system. The behavior of the quantum system may be assumed to be independent of the nature of the thermostat. The most convenient object of the quantum theory to describe such non-isolated systems is the density matrix since the partition function does not determine any local thermodynamic quantities. It is for this reason that we introduced quantum mechanical and quantum statistical density matrices which are necessary to do the investigation via path integration. This enables us to make a transition to the quasi-classical approximation of quantum statistical mechanics, which, in turn, enables us to evaluate the first quantum correction to the classical Boltzmann distribution function.
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.
Overview Classical Statistical Mechanics (CSM) ignores the quantum nature of particles. This introduces three errors in the calculation of the partition function. First is the assumption of distinguishability of individual particles not possible in the microscopic world. Second, CSM neglects the discrete nature of system’s energy levels. Third, CSM fails to account for the spin of the particles which divides all particles into two types: boson and fermion, each following its own law of statistical distribution among energy levels. While CSM is impressively successful in describing observed properties of many naturally occurring large scale phenomena (particularly in liquids and biological systems), it fails to describe many phenomena where quantum nature of particles and quantum statistics play important role in determining properties such as electrical and thermal conductivity (particularly in solid materials) and superfluidity in liquid helium. It is not just at low temperatures where quantum nature is important, but also when the particle is intrinsically quantum, such as an electron. Quantum Statistical Mechanics (QSM) is also needed to understand biochemical reactions, like enzyme kinetics and optical properties of DNA. In this chapter, we discuss elementary aspects of quantum Statistical Mechanics, with detailed treatment of ideal Bose gas, leading to Bose-Einstein condensation.
Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)
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.