Schrodingers equation (SE) and the information-optimizing principle based on Fishers 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 states 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.
It is well known that a suggestive connection links Schrodingers equation (SE) and the information-optimizing principle based on Fishers 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 SEs 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 Schrodingers equation. Remarkably enough, and in contrast with standard variational approaches, our present procedure does not involve free fitting parameters.
It is well known that a suggestive relation exists that links Schrodingers equation (SE) to the information-optimizing principle based on Fishers 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 SEs 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.
By recourse to i) the Hellmann-Feynman theorem and ii) the Virial one, the information-optimizing principle based on Fishers information measure uncovers a Legendre-transform structure associated with Schrodingers 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 Fishers measure that exist between Schrodingers equation, on the one hand, and thermodynamics/thermostatistics on the other one.
It is well known that a suggestive relation exists that links Schrodingers equation (SE) to the information-optimizing principle based on Fishers 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.
