Direct Fisher Inference of the Quartic Oscillator's Eigenvalues

S. P. Flego, Angelo Plastino, A. R. Plastino
2011 Journal of Modern Physics  
It is well known that a suggestive connection links Schr\"odinger'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
more » ... f 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 Schr\"odinger's equation. Remarkably enough, and in contrast with standard variational approaches, our present procedure does not involve free fitting parameters.
doi:10.4236/jmp.2011.211171 fatcat:jk4fdmoiendnlhqbe62zoszfwq