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.
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.
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.
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. 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 the 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.
A suggestive relation links Fisher' information measure (FIM)Iand Schrödinger equation (SE). The connection is based upon the fact that the constrained minimization of I leads to a SE. This, in turn, is the origin of intriguing relationships between various aspects of SE, on the one hand, and the formalism of statistical mechanics derived from Jaynes's maximum entropy principle (MaxEnt), on the other one. The link entails the existence of a Legendre transform structure underlying the SE, which allows for the emergence of two first-order differential equations that must, respectively, be satisfied by i) the Fisher measure and ii) the SE energy eigenvalues. The complete A) I-solution and B) energy-solution are both obtained bypassing the SE and, furthermore, linked by the Legendre structure.
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.
Abstract We study the scaling properties of Fisher’s information measure (FIM) and show that from these one can straightforwardly deduce significant quantum-mechanical results. Specifically, we investigate the scaling properties of Fisher’s measure I and encounter that, from the concomitant operating rules, several interesting, albeit known, results can be derived. This entails that such results can be regarded as pre-configured by the conjunction of scaling and information theory. The central notion to be arrived at is that scaling entails that I must obey a certain partial differential equation (PDE). These PDE-solutions have properties that enable the application of a Legendre-transform (LT). The conjunction PDE+LT leads one to obtain several quantum results without recourse to the Schrödinger’s equation.
In (1) the Schrodinger equation is derived using the extremization of Fisher information in a certain way formulated in (2) i.e. Extreme Physical Information. Fisher information, however, is defined in terms of P(x) which in quantum mechanics is W*(x)W(x) where W(x) is the wavefunction. As a result, the Schrodinger equation is derived in terms of P(x) and then the mathematical substitution P(x)=W(x)W(x) for a bound state “simplifies” the form of the equation. Otherwise there would be no need to introduce the wavefunction. Furthermore P(x) = constant for a quantum free particle which is also a solution of the Schrodinger equation. For P(x)=constant, however, one would normally not consider using the Fisher information. We further note that in a physical example, namely that of the reflection/refraction of light from an interface of media one is forced to introduce the concept of a wavefunction i.e. the square root of flux, otherwise the problem cannot be solved. Thus in this situation one does not solve the problem in terms of P(x) and then introduce W(x) to simplify the form. In particular, if one considers a single photon moving to the right, it either reflects or refracts (with probability), but the ensuing equations involve the weight of the square root fluxes (wavefunctions) of all three. How can this be unless there is uncertainty in time and space which allows one to have all three together? Thus the reflection/refraction problem depends on square root probability or flux and introduces the idea of uncertainty in x, t. Given the translational invariance in the problem, it is natural to introduce the translation operator d/dx. In fact, one may define the uncertainty in x such that one has an eigenfunction of -id/dx i.e. exp(ipx). This is complex and so its modulus respects P(x)=constant which would yield a Fisher’s information of 0, meaning that there is no probability and hence no need for Fisher’s information. There is, however, probability in x, but a different level than the deterministic x=vt where v=constant. Thus we argue that one may obtain exp(ipx) strictly from the notion of translational invariance. In other words, Fisher information is not required to find this form and the quantum physics associated with it. Nevertheless Fisher information and EPI (Extreme physical Information) may be used to derive the Schrodinger equation (applied to a bound state) and one may deduce exp(ipx) from this. We note, however, that Fisher information makes use of d/dx which is the translational generator and so the idea of translational invariance and Fisher’s information/EPI may be linked.