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.
Highlights
It is well-known that a strong link exists between Fisher’ information measure (FIM) I [1] and Schrödinger wave equation (SE) [2,3,4,5,6,7]
Basic s equation (SE)-consequences such as the Hellmann-Feynman and the Virial theorems can be reinterpreted in terms of a special kind of reciprocity relations between relevant physical quantities, similar to the ones exhibited by the thermodynamics’ formalism via its Legendre-invariance property [5,6]
Without loss of generality, renormalize the reference quantities Fk. This procedure is convenient because it allows us to regard these quantities as statistical weights that optimize the Cramer Rao (CR)-bound
Summary
It is well-known that a strong link exists between Fisher’ information measure (FIM) I [1] and Schrödinger wave equation (SE) [2,3,4,5,6,7] In a nutshell, this connection is based upon the fact that a constrained Fisher-minimizetion leads to a SE-like equation [1,2,3,4,5,6,7]. Basic SE-consequences such as the Hellmann-Feynman and the Virial theorems can be reinterpreted in terms of a special kind of reciprocity relations between relevant physical quantities, similar to the ones exhibited by the thermodynamics’ formalism via its Legendre-invariance property [5,6] This fact demonstrates that a Legendre-transform structure underlies the non-relativistic Schrödinger equation. The parameter-free nature of our treatment is evidence of the structural physical information that we incorporate to the theory via Fisher's information measure
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