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Direct Fisher Inference of the Quartic Oscillator’s Eigenvalues

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Author(s): S. P. Flego | Angelo Plastino | A. R. Plastino

Journal: Journal of Modern Physics
ISSN 2153-1196

Volume: 02;
Issue: 11;
Start page: 1390;
Date: 2011;
Original page

Keywords: Information Theory | Fisher’s Information Measure | Legendre Transform | Quartic Anharmonic Oscillator

ABSTRACT
It is well known that a suggestive connection links Schrödinger’s equation (SE) and the information-optimizing principle based on Fisher’s information measure (FIM). It has been shown that this entails the existence of a Legendre transform structure underlying the SE. Such a structure leads to a first order partial differential equation (PDE) for the SE’s eigenvalues from which a complete solution for them can be obtained. We test this theory with regards to anharmonic oscillators (AHO). AHO pose a long-standing problem and received intense attention motivated by problems in quantum field theory and molecular physics. By appeal to the Cramer Rao bound we are able to Fisher-infer the energy eigenvalues without explicitly solving Schrödinger’s equation. Remarkably enough, and in contrast with standard variational approaches, our present procedure does not involve free fitting parameters.
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