Statistical Mechanics · Problem 8.21

Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions — R. K. Pathria

Physics — Statistical Mechanics · thomas-fermi-atom-virial-theorem

Problem

The total energy

E

of the electron cloud in an atom can be written as

E = K + V_{ne} + V_{ee},

where

K

is the kinetic energy of the electrons,

V_{ne}

the interaction energy between the electrons and the nucleus, and

V_{ee}

the mutual interaction energy of the electrons. Show that, according to the Thomas--Fermi model of a neutral atom,

K = -E, \quad V_{ne} = +\tfrac{7}{3}E \quad \text{and} \quad V_{ee} = -\tfrac{1}{3}E,

so that total

V(= V_{ne} + V_{ee}) = 2E

. Note that these results are consistent with the virial theorem; see Problem 3.20, with

n = -1