  Variational extremization of the relative Fisher information (RFI, hereafter)
is performed. Reciprocity relations, akin to those of thermodynamics are
derived, employing the extremal results of the RFI expressed in terms of
probability amplitudes. A time independent Schr\"{o}dinger-like equation
(Schr\"{o}dinger-like link) for the RFI is derived. The concomitant Legendre
transform structure (LTS, hereafter) is developed by utilizing a generalized
RFI-Euler theorem, which shows that the entire mathematical structure of
thermodynamics translates into the RFI framework, both for equilibrium and
non-equilibrium cases. The qualitatively distinct nature of the present results
\textit{vis-\'{a}-vis} those of prior studies utilizing the Shannon entropy
and/or the Fisher information measure (FIM, hereafter) is discussed. A
principled relationship between the RFI and the FIM frameworks is derived. The
utility of this relationship is demonstrated by an example wherein the energy
eigenvalues of the Schr\"{o}dinger-like link for the RFI is inferred solely
using the quantum mechanical virial theorem and the LTS of the RFI.
