Abstract This chapter develops the stability theory needed for the existence of the classical partition. This is extended to quantum stability with emphasis on the physically important case of nonrelativistic quantum mechanics with Coulomb interactions. The existence and uniqueness of the thermodynamic limit is discussed in detail. Open questions are presented.
Abstract This chapter develops the stability theory needed for the existence of the classical partition. This is extended to quantum stability with emphasis on the physically important case of nonrelativistic quantum mechanics with Coulomb interactions. The existence and uniqueness of the thermodynamic limit is discussed in detail. Open questions are presented.
Abstract This chapter opens with the notion of stable interaction in quantum systems, somewhat different from classical systems. It is shown that an unstable classical system can become stable when treated quantum mechanically. The remaining of the chapter is devoted to a brief survey of the main results concerning the stability of Coulomb systems. Explicit evaluations of the total energy show that the exclusion principle is essential for their stability. The final part of the chapter deals with the thermodynamic limit of globally neutral Coulomb systems. The limit exists in spite of the long-range character of the interaction, and contrary to the stability property, the existence of the limit does not require quantum mechanics, being valid also in classical Coulomb systems.
We present recent works [15,16] on the thermodynamic limit of quantum Coulomb systems. We provide a general method which allows to show the existence of the limit for many different systems. Ordinary matter is composed of electrons (negatively charged) and nuclei (positively charged) interacting via Coulomb forces. The potential between two particles of charges z and z ′ located at x and x ′ in R 3 is zz ′ |x − x ′ |. There are two difficulties which occur when trying to describe systems composed of electrons and nuclei. Both have to do with the physical problem Talk given by M.L. at QMath10, 10th Quantum Mathematics International Conference, Moeciu, Romania, September 10–15, 2007. c○2008 by the authors. This work may be reproduced, in its entirety, for non-commercial purposes. 2 of stability of quantum systems. The first is due to the singularity of 1/|x | at 0: it is necessary to explain why a particle will not rush to a particle of the opposite charge. One of the first major triumphs of the theory of quantum mechanics is the explanation it gives of the stability of the hydrogen atom (and the complete description of its spectrum) and of other microscopic quantum Coulomb systems, via the uncertainty principle. Stability means that the total energy of the considered system cannot be arbitrarily negative. If there was no such lower bound to the energy it would be possible in principle to extract an infinite amount of energy. One often refers to this kind of stability as stability of the first kind [19,20]. If we denote by E(N) the ground state energy of the system under consideration, for N particles stability of the first kind can be written E(N)> −∞. (1) In proving (1) for Coulomb systems, a major role is played by the uncertainty principle which for nonrelativistic systems is mathematically expressed by the critical Sobolev embedding H 1 (R 3) ֒ → L 6 (R 3). The latter allows to prove Kato’s inequality
We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the ε j 's and g j 's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics. We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.
QM∞ includes quantum field theory and the thermodynamic limit of quantum statistical mechanics — theories which, unlike the 'ordinary' quantum theories typically discussed by philosophers, concern infinite systems. The mathematical framework appropriate for the presentation of a theory of ordinary quantum mechanics is essentially unique. For theories of QM∞, this is not so. This prompts interpretive questions—for instance, about what makes a quantum theory the quantum theory it is — this work aims to chronicle. Having introduced a formal apparatus (operator algebra theory) suited to pursue these questions, the book articulates a variety of accounts of the content of quantum theories, accounts responsive to QM∞'s characteristic non-uniqueness. To evaluate these accounts, the book examines QM∞ settings (e.g. spontaneous symmetry breaking, cosmological particle creation, superconductivity) in which that characteristic non-uniqueness seems to matter, with a view toward determining which accounts sustain the uses to which the non-uniqueness is put. This approach not only brings work on the foundations of quantum theories in contact with the foundational investigation of other sorts of physical theories (e.g., thermodynamics, statistical mechanics, solid state physics, general relativity, and cosmology), it also brings the philosophy of physics in contact with other sorts of philosophy of science (e.g., accounts of explanation, reduction, and explanationist defenses of scientific realism). The book concludes that received notions of physical content and physical modality must be revised if they are to apply usefully to particular physical theories.
When quantum theory is applied to systems with an infinite number of degrees of freedom, wholly new interpretative issues arise. This is due to the fact that such theories have physically inequivalent instantiations. Here the interpretative issues of quantum field theory and of the quantum theory of systems at the thermodynamic limit are discussed. In quantum mechanics the so-called commutation operators on the basic descriptive operators fix the descriptive realm of the theory. This chapter addresses the question of how we are to understand this theory that seems to allow for distinct, incompatible ways of representing the basic features of the world.