The purpose of this article is to show that electrons and protons, interacting by Coulomb forces and governed by quantum statistical mechanics at suitable temperature and density, form a gas of Hydrogen atoms or molecules.
In this paper we address a question posed by M. and T. Hoffmann-Ostenhof, which concerns the total spin of the ground state of an atom or molecule. Each electron is given a value for spin, ±1/2. The total spin is the sum of the individual spins.
In [FS1] we announced a precise asymptotic formula for the ground-state energy of a non-relativistic atom. The purpose of this paper is to establish an elementary inequality that plays a crucial role in our proof of that formula. The inequality concerns the Thomas-Fermi potential VTF = -y(ar) / r, a > 0, where y(r) is defined as the solution of ⎧ y''(x) = x-1/2y3/2(x), ⎨ y(0) =...
We review recent work of the authors on the non-relativistic Schrödinger equation with a honeycomb lattice potential, . In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution , for data , which is spectrally localized near a Dirac point. We conclude with a formal derivation and discussion of the...
We extend Van der Corput's method for exponential sums to study an oscillating term appearing in the quantum theory of large atoms. We obtain an interpretation in terms of classical dynamics and we produce sharp asymptotic upper and lower bounds for the oscillations.
We establish a Lieb-Thirring type estimate for Pauli Hamiltonians with non-homogeneous magnetic fields. Besides of depending on the size of the field, the bound also takes into account the size of the field gradient. We then apply the inequality to prove stability of non-relativistic quantum mechanical matter coupled to the quantized ultraviolet-cutoff electromagnetic field for arbitrary values of the fine structure constant.
In this article, we study the quantum mechanics of N electrons and M nuclei interacting by Coulomb forces. Motivated by an important idea of Chandrasekhar and following Herbst [H], we modify the usual kinetic energy -∆ to take into account an effect from special relativity. As a result, the system can implode for unfavorable values of the nuclear charge Z and the fine structure constant α. This is analogous to the gravitational collapse of a heavy star. Our goal here is to find those values of α...
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