Fisher information, the Hellmann–Feynman theorem, and the Jaynes reciprocity relations

Flego, Silvana; Plastino, A.

doi:10.1016/j.aop.2011.07.009

Cited by 17 publications

(31 citation statements)

References 29 publications

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“…While relations analogous to the one between the Lagrange multipliers and the coefficients of the QM potential inferred from empirical observables (18) have been derived in [18,22], Ref. [18] makes no attempt to infer the energy-eigenvalues and obtain the pdf profiles.…”

Section: Discussionmentioning

confidence: 99%

“…where a k are coefficients related to the concomitant Lagrange multipliers through (18). An observation of much significance arises here: while the energy-eigenvalues and the concomitant pdf profiles pertaining to the RFI framework noticeably differ from their FIM counterparts, the Lagrange multipliers for the constraint terms, corresponding to the experimental observables, are readily demonstrated to be the same for both the RFI and the FIM scenarios.…”

Section: Analysis and Numerical Simulationsmentioning

confidence: 92%

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Hellmann–Feynman connection for the relative Fisher information

Venkatesan¹,

Plastino

2015

Annals of Physics

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h i g h l i g h t s• Legendre transform structure for the RFI is obtained with the Hellmann-Feynman theorem.• Inference of the energy-eigenvalues of the SWE-like equation for the RFI is accomplished.• Basis for reconstruction of the RFI framework from the FIM-case is established.• Substantial qualitative and quantitative distinctions with prior studies are discussed. a b s t r a c tThe (i) reciprocity relations for the relative Fisher information (RFI, hereafter) and (ii) a generalized RFI-Euler theorem are selfconsistently 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 Schrödinger-like equation can be associated to the RFI. Lagrange multipliers are determined invoking the RFI-LTS link and the quantum mechanical virial theorem. An appropriate ansatz allows for the inference of probability density functions (pdf's, hereafter) and energy-eigenvalues of the above mentioned Schrödinger-like equation. The energyeigenvalues obtained here via inference are benchmarked against 301 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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Section: Discussionmentioning

confidence: 99%

Section: Analysis and Numerical Simulationsmentioning

confidence: 92%

Hellmann–Feynman connection for the relative Fisher information

Venkatesan¹,

Plastino

2015

Annals of Physics

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“…Appeal to the Hellmann-Feynman theorem is now needed here with reference to Cauchy-uniqueness. For our present case, the Lipschitz condition can be seen to be verified always since the Hellmann-Feynman theorem [3][4][5] guarantees that…”

Section: Interpreting Equation (13)mentioning

confidence: 99%

“…The theorem has been proven independently by many authors, including Guettinger (1932), Pauli (1933), Hellmann (1937), and Feynman (1939) [3][4][5].…”

Section: The Hellmann-feynman Rheoremmentioning

confidence: 99%