The total energy \(E\) of the electron cloud in an atom can be written as

\[
E=K+V_{n e}+V_{e e},
\]

where \(K\) is the kinetic energy of the electrons, \(V_{n e}\) the interaction energy between the electrons and the nucleus, and \(V_{e e}\) the mutual interaction energy of the electrons. Show that, according to the Thomas-Fermi model of a neutral atom,

\[
K=-E, \quad V_{n e}=+\frac{7}{3} E, \quad \text { and } \quad V_{e e}=-\frac{1}{3} E
\]

so that total \(V=V_{n e}+V_{e e}=2 E\). Note that these results are consistent with the virial theorem; see Problem 3.20, with \(n=-1\).