Fluctuations of collective coordinates and convexity theorems for energy surfaces

TL;DR

This paper explores the relationship between fluctuations of collective coordinates and the convexity of energy surfaces in many-body systems, emphasizing the importance of considering uncertainties in both energy and collective variables for accurate modeling.

Contribution

It introduces the significance of fluctuations in collective coordinates and establishes a connection between these fluctuations and the convexity properties of energy surfaces.

Findings

Fluctuations of collective coordinates are crucial for understanding energy surface properties.

Convexity theorems relate the shape of energy surfaces to collective coordinate fluctuations.

Considering uncertainties in both energy and collective variables improves theoretical models.

Abstract

Constrained energy minimizations of a many-body Hamiltonian return energy landscapes e(b) where b=<B> representes the average value(s) of one (or several) collective operator(s), B, in an "optimized" trial state Phi_b, and e = <H> is the average value of the Hamiltonian in this state Phi_b. It is natural to consider the uncertainty, Delta e, given that Phi_b usually belongs to a restricted set of trial states. However, we demonstrate that the uncertainty, Delta b, 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.