Parameter-free ansatz for inferring ground state wave functions of even potentials

Physica Scripta (Impact Factor: 1.13). 07/2011; 85(5). DOI: 10.1088/0031-8949/85/05/055002
Source: arXiv


Schr\"odinger's equation (SE) and the information-optimizing principle based
on Fisher's information measure (FIM) are intimately linked, which entails the
existence of a Legendre transform structure underlying the SE. In this
comunication we show that the existence of such an structure allows, via the
virial theorem, for the formulation of a parameter-free ground state's
SE-ansatz for a rather large family of potentials. The parameter-free nature of
the ansatz derives from the structural information it incorporates through its
Legendre properties.

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    ABSTRACT: The $(i)$ reciprocity relations for the relative Fisher information (RFI, hereafter) and $(ii)$ a generalized RFI-Euler theorem, are self-consistently derived from the Hellmann-Feynman theorem. These new reciprocity relations generalize the RFI-Euler theorem and constitute the basis for building up a mathematical Legendre transform structure (LTS, hereafter), akin to that of thermodynamics, that underlies the RFI scenario. This demonstrates the possibility of translating the entire mathematical structure of thermodynamics into a RFI-based theoretical framework. Virial theorems play a prominent role in this endeavor, as a Schr\"odinger-like equation can be associated to the RFI. Lagrange multipliers are determined invoking the RFI-LTS link and the quantum mechanical (QM) virial theorem. An appropriate ansatz allows for the inference of probability density functions (pdf's, hereafter) and energy-eigenvalues of the above mentioned Schr\"odinger-like equation. The energy-eigenvalues obtained here via inference are benchmarked against established theoretical and numerical results. A principled theoretical basis to reconstruct the RFI-framework from the FIM framework is established. Numerical examples for exemplary cases are provided.
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