Hypervirial relations between ground-state particle density, kinetic energy density, and potential in three dimensions

Abstract

Previous work on one-dimensional hypervirial theorems (HVT) is extended to three dimensions. For a restricted class of hypervirial operators A ̂ relations valid for single stationary states can be summed up to yield exact relations between ϱ (ground-state particle density) and ϵ k (kinetic energy density) of a system of fermions moving independently in a common one-body potential V( r). For the type A ̂ = Σ α=1 3 \\ ̂ tf α( r) p ̂ α the most general vector field ‖ α is determined which allows for summation over states yielding, besides results already known, a new HVT. Another new HVT connecting ϱ and ϵ k is derived from A ̂ = p ̂ α Σ β=1 3 p ̂ β 2 . The Thomas-Fermi expressions for ϱ and ϵ k are testedd whether they satisfy these relations.