arXiv subjects

S. P. Flego

Publications and source records attributed to S. P. Flego.

7 recordsLinked to original sources

arXiv2021

Virial ansätze for the Schrödinger Equation with a symmetric strictly convex potential. Part II

Recently was introduced in the literature a procedure to obtain ansätze, free of parameters, for the eigenfunctions of the time-independent Schrödinger equation with symmetric convex potential. In the present work, we test this technique in regard to $x^{2κ}$-type potentials. We study the behavior of the ansätze regarding the degree of the potential and to the intervening coupling constant. Finally, we discuss how the results could be used to establish the upper bounds of the relative errors in situations where intervening polynomial potentials.

quant-ph↗

arXiv2020

Virial-ansätze for the Schrödinger Equation with a symmetric strictly convex potential

Considering symmetric strictly convex potentials, a local relationship is inferred from the virial theorem, based on which a real log-concave function can be constructed. Using this as a weight function and in such a way that the virial theorem can still be verified, parameter-free ansätze for the eigenfunctions of the associated Schrödinger equation are built. To illustrate the process, the technique is successfully tested against the harmonic oscillator, in which it leads to the exact eigenfunctions, and against the quartic anharmonic oscillator, which is considered the paradigmatic testing ground for new approaches to the Schrödinger equation.

math-ph↗

arXiv2011

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

Schrödinger'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.

quant-ph↗

arXiv2011

Direct Fisher inference of the quartic oscillator's eigenvalues

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. As an application we deal with the quantum theory of anharmonic oscillators, a long-standing problem that has 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 particular PDE-solution that yields the 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.

quant-ph↗

arXiv2011

Special features of the relation between Fisher Information and Schrödinger eigenvalue equation

It is well known that a suggestive relation exists that links Schrödinger's equation (SE) to the information-optimizing principle based on Fisher's information measure (FIM). The connection entails the existence of a Legendre transform structure underlying the SE. Here we show that appeal to this structure leads to a first order differential equation for the SE's eigenvalues that, in certain cases, can be used to obtain the eigenvalues without explicitly solving SE. Complying with the above mentioned equation constitutes a necessary condition to be satisfied by an energy eigenvalue. We show that the general solution is unique.

quant-ph↗

arXiv2011

Inferring an optimal Fisher measure

It is well known that a suggestive relation exists that links Schrödinger's equation (SE) to the information-optimizing principle based on Fisher's information measure (FIM). We explore here an approach that will allow one to infer the optimal FIM compatible with a given amount of prior information without explicitly solving first the associated SE. This technique is based on the virial theorem and it provides analytic solutions for the physically relevant FIM, that which is minimal subject to the constraints posed by the prior information.

math.ST↗

arXiv2011

Legendre-transform structure derived from quantum theorems

By recourse to i) the Hellmann-Feynman theorem and ii) the Virial one, the information-optimizing principle based on Fisher's information measure uncovers a Legendre-transform structure associated with Schrödinger's equation, in close analogy with the structure that lies behind the standard thermodynamical formalism. The present developments provide new evidence for the information theoretical links based on Fisher's measure that exist between Schrödinger's equation, on the one hand, and thermodynamics/thermostatistics on the other one.

cond-mat.stat-mech↗