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Quantum Long-Range Systems

Alessandro Campa, Thierry Dauxois, Duccio Fanelli, Stefano Ruffo

Published Aug 28, 2014• 0 citations
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780199581931.003.0006

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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 study the thermodynamics of quantum particles with long-range interactions at T = 0. Specifically, we generalize the Hamiltonian mean-field (HMF) model to the case of bosons. We consider the Hartree approximation that becomes exact in a proper thermodynamic limit with a coupling constant k ∼ 1/ N . The equilibrium configurations are solutions of the mean-field Schrödinger equation with a cosine interaction. We show that the homogeneous phase, which is unstable in the classical regime, becomes stable in the quantum regime. The homogeneous phase is stabilized by the Heisenberg uncertainty principle. This takes place through a second-order phase transition where the control parameter is the normalized Planck constant. The homogeneous phase is unstable for and stable for . The inhomogeneous phase is stable for and disappears for . We point out analogies between the bosonic HMF model and the concept of boson stars in astrophysics. We also discuss the differences between bosons and fermions for what concerns the thermodynamic limit, the order of the phase transition and the form of the density profiles.
This chapter attempts to expound on basic and essential ideas (for further use in the book) from both classical and quantum mechanics. The chapter is somewhat technical in nature but only requires an elemental knowledge of calculus. The first three sections take a review of some of the elements of classical mechanics and classical statistical mechanics – the Euler–Lagrange and the Hamilton–Jacobi equations, the idea of an ensemble in the classical context, and the continuity equation for particle density. The remaining part is devoted to the elements of quantum mechanics – the connection between the Hamilton–Jacobi equation and the Schrödinger equation, the idea of an ensemble in the quantum context, the free particle wave function and operators, the uncertainty principle and the idea of the expectation value of an operator, and the concept of a wave packet.
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