Direct Fisher Inference of the Quartic Oscillator’s Eigenvalues

S. P. Flego, Angelo Plastino, A. R. Plastino

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DOI: 10.4236/jmp.2011.211171                

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.

Keywords

Information Theory, Fisher’s Information Measure, Legendre Transform, Quartic Anharmonic Oscillator

Conflicts of Interest

The authors declare no conflicts of interest.

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