Fluctuations of collective coordinates and convexity theorems for energy surfaces

Constrained energy minimizations of a many-body Hamiltonian return energy landscapes e(b) where b≡〈B〉 represents the average value(s) of one (or several) collective operator(s), B, in an “optimized” trial state Φ_b, and e≡〈H〉 is the average value of the Hamiltonian in this state Φ_b. It is natural to consider the uncertainty, Δe, given that Φ_b usually belongs to a restricted set of trial states. However, we demonstrate that the uncertainty, Δb, must also be considered, acknowledging corrections to theoretical models. We also find a link between fluctuations of collective coordinates and convexity properties of energy surfaces.