Statistical theory of the atom in momentum space

Abstract

In 1992, Englert [Phys. Rev. A, 45;127--134] found a momentum energy functional for atoms and discussed the relation to the Thomas-Fermi functional (Lenz [Z. Phys., 77;713--721]). We place this model in a mathematical setting. Our results include a proof of existence and uniqueness of a minimizing momentum density for this momentum energy functional. Further, we investigate some properties of this minimizer, among them the connection with Euler's equation. We relate the minimizers of the Thomas-Fermi functional and the momentum energy functional found by Englert by explicit transforms. It turns out that in this way results well-known in the Thomas-Fermi model can be transferred directly to the model under consideration. In fact, we gain equivalence of the two functionals upon minimization. Finally, we consider momentum dependent perturbations. In particular, we show that the atomic momentum density converges to the minimizer of the momentum energy functional as the total nuclear charge becomes large in a certain sense. This thesis is based on joint work with Prof. Dr. Heinz Siedentop and the main contents will also appear in a joint article