On alternative forms of the variational principle implying the generalized Thomas-Fermi theory

A gas of fermions in a static potential is investigated within the framework of the March-Murray perturbation scheme equivalent to a generalization of the Thomas-Fermi theory. The corresponding variational principle, due to Stoddart and March, is complemented by an alternative form. In contrast with the Stoddart-March formulation whose nature is Hamiltonian and which requires variations of the particle density in the total energy functional, the author's formulation, found to be of Lagrangian nature, does variations with respect to the potential function. Both the (alternative) quantum variational (many-body) principles are formulated self-consistently, with due attention to the Poisson equation of electrodynamics. They are considered first for zero temperature and afterwards for arbitrary temperatures. The functional considerations involved in the present paper are related with thermodynamics of quantum microcontinuum. An Appendix is added, to clarify the functional concepts by demonstrating them on some simple analogy. Therefore, a variational calculation of the ground state energy of ^– mesoatoms or mesoions is given. It leads mathematically to the well-known problem of calculating the ground 1 S (parastate) energy of two electrons in the helium atom.