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تسجيل مستخدم جديدDirect Fisher inference of the quartic oscillators eigenvalues
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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.
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We introduce a nilpotent group to write a generalized quartic anharmonic oscillator Hamiltonian as a polynomial in the generators of the group. Energy eigenvalues are then seen to depend on the values of the two Casimir operators of the group. This d
ependence exhibits a scaling law which follows from the scaling properties of the group generators. Demanding that the potential give rise to polynomial solutions in a particular Lie algebra element puts constraints on the four potential parameters, leaving only two of them free. For potentials satisfying such constraints at least one of the energy eigenvalues and the corresponding eigenfunctions can be obtained in closed analytic form; examples, beyond those available in the literature, are given. Finally, we find solutions for particles in external electromagnetic fields in terms of these quartic polynomial solutions.
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Consider two $p$-variate populations, not necessarily Gaussian, with covariance matrices $Sigma_1$ and $Sigma_2$, respectively, and let $S_1$ and $S_2$ be the sample covariances matrices from samples of the populations with degrees of freedom $T$ and
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