Fisher information, the Hellmann–Feynman theorem, and the Jaynes reciprocity relations

We explore intriguing links connecting Hellmann–Feynman’s theorem to a thermodynamics information-optimizing principle based on Fisher’s information measure.

► We link a purely quantum mechanical result, the Hellmann–Feynman theorem, with Jaynes’ information theoretical reciprocity relations.
► These relations involve the coefficients of a series expansion of the potential function.
► We suggest the existence of a Legendre transform structure behind Schrödinger’s equation, akin to the one characterizing thermodynamics.