Fisher Information and Quantum Mechanics
S. P. Flego *
Universidad Nacional de La Plata, Facultad de Ingeniería, Grupo de Investigación Teórica y Aplicada de la Teoría de la Información (GTyATI), 1900 La Plata, Argentina
A. Plastino
Universidad Nacional de La Plata, Instituto de Física (IFLP-CCT-CONICET), C.C. 727, 1900 La Plata, Argentina and Instituto Carlos I de Física Teórica y Computacional and Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada, Spain and Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca, Spain
A. R. Plastino
CREG-Universidad Nacional de La Plata-CONICET, C.C. 727, 1900 La Plata, Argentina and Instituto Carlos I de Física Teórica y Computacional and Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada, Spain
*Author to whom correspondence should be addressed.
Abstract
A suggestive relation links Fisher' information measure (FIM)Iand Schrödinger equation (SE). The connection is based upon the fact that the constrained minimization of I leads to a SE. This, in turn, is the origin of intriguing relationships between various aspects of SE, on the one hand, and the formalism of statistical mechanics derived from Jaynes's maximum entropy principle (MaxEnt), on the other one. The link entails the existence of a Legendre transform structure underlying the SE, which allows for the emergence of two first-order differential equations that must, respectively, be satisfied by i) the Fisher measure and ii) the SE energy eigenvalues. The complete A) I-solution and B) energy-solution are both obtained bypassing the SE and, furthermore, linked by the Legendre structure.
Keywords: Fisher information, MaxEnt, legendre structure, reciprocity relations, Virial theorem, Hellmann-Feynman theorem