Pedagogic notes on Thomas-Fermi theory (and on some improvements): atoms, stars, and the stability of bulk matter

In the more than half century since the semiclassical Thomas-Fermi theory of the atom was introduced, there have been literally thousands of publications based on that theory; they encompass a broad range of atomic bound-state and scattering problems. (The theory has also been applied to nuclear physics and solid-state problems.) We will concentrate here on the essence of the theory, namely, its implementation of the uncertainty and exclusion principles and of the Coulomb or Newton force law. Since we are often far more interested in physical concepts than in numerical accuracy or rigor, we will sometimes consider the implementation in a qualitative rather than quantitative fashion. The theory is then capable of giving only qualitative information about a system—one obtains the dependence of the total ground-state binding energy $E$ and radius $R$ of an atom on the nuclear charge $Z$, for example, but one obtains only rough estimates of the numerical coefficients; in compensation, the calculations are often literally trivial, very much simpler than the already simple Thomas-Fermi calculations. A point to be emphasized is that in the course of obtaining an estimate of $E$ and $R$ of an atom in a Thomas-Fermi approach, one also obtains an estimate of the electronic density, and, particularly if the analysis is more than simply qualitative, an electronic-density estimate can be very useful in a wide variety of problems. We include a short comment on alternative formulations of Thomas-Fermi theory in a $D$-dimensional space. We will review the applications of the theory, from both qualitative and (Thomas-Fermi) quantitative viewpoints, to heavy atoms, where we are concerned with a Coulomb interaction, and to neutron stars and white dwarfs, where we are concerned with a gravitational interaction and with gravitational-plus-Coulomb interactions, respectively. In the latter case, the first two Coulomb corrections are evaluated. Very rough (relativistic) estimates are made of the conditions under which heavy atoms, neutron stars, and white dwarfs collapse. A one-dimensional Thomas-Fermi-like theory also exists for heavy atoms in a uniform strong magnetic field $B$, of the order of the field believed to exist at the surface of a neutron star. Here, too, the qualitative picture immediately gives some of the main results, namely, the dependence of $E$ and $R$ on $B$ and $Z$. We also comment briefly on some relatively recent and very recent developments in Thomas-Fermi theory. These include a proof of the stability of matter. Though it was first proved by Dyson and Lenard, we consider the Lieb-Thirring proof, both because it is much simpler and because it makes extensive use of Thomas-Fermi theory, including a no-molecular-binding theorem that follows in the Thomas-Fermi approximation: Teller proved that, in that approximation, atoms could not form molecular bound states. These developments also include (a) the Lieb-Simon proof that the prediction of the theory that $E=-c_7{Z}^{\frac{7}{3}}$, with $c_7$ a specified coefficient, becomes exact at $Z\sim \infty$, (b) the Scott-$c_6{Z}^{\frac{6}{3}}$ correction term, with $c_6$ specified and now known also to be exact, and (c) the Schwinger estimate of the coefficient $c_5$ of the $Z^{\frac{5}{3}}$ term, which there is good reason to believe is exact. The many digressions include comments on QED, on lower bounds on the ground-state energy of a system, and on mini-boson stars.