Thomas-Fermi Theory of the Atom as a Solution of the Density-Matrix Hierarchy

In this paper we show that the Thomas-Fermi (TF) theory for a neutral atom is the zeroth-order solution of the full $N$-body problem if (a) the following assignments of smallness are made: $N^{-\frac{1}{3}}=\epsilon$, $e=e_0{\epsilon }^{\frac{3}{2}}$, $m=\frac{m_0}{{\epsilon }^2}$, $\epsilon \to 0$; (b) the singlet density matrix in the $x$ representation, ${\rho }_1(x^{\prime }, x)$, is a fast oscillating function of the off-diagonal elements, of the type $sin\left[\frac{(x^{\prime }-x)}{\epsilon }\right]$; (c) the higher-order density matrices are determinants of the singlet density matrix as $\epsilon \to 0$. The higher-order approximations, however, cannot be obtained by a simple power-series expansion in $\epsilon$, since the solutions contain $\epsilon$ in a nonanalytical fashion. Taking into account the exclusion principle to zeroth order in $\epsilon$, and solving the equations of motion for the singlet density matrix to the next-higher-(second-) order approximation, we obtain the equations found by Kompaneets and Pavlovskii, and Baraff and Borowitz. These contain the Dirac and Weizsäcker corrections. Finally, we offer some conjectures about possible improvements of the approximation scheme.