1 The Thermodynamic Limit of Quantum Coulomb Systems: A New Approach
Christian Hainzl, Mathieu Lewin B, Jan Philip Solovej
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
•Jan 1, 2008•2 citations