Applying Lucas Succession to the Harmonic Hamiltonian of the Representation of Born and Jordan

The analysis is carried out using Lucas Sequence considering a vector space of finite dimension, in which the action of a linear operator on any vector in the space will be well defined as long as its action on each of the vectors of a base is well defined. defined. This however cannot be true in the case of a finite dimensional space, two examples are given using the matrix representation of the Hamiltonian of the harmonic oscillator written in (1.24), constructed in terms of the orthonormal basis (infinite but enumerable) formed by the eigenvectors of the Hamiltonian [1], presenting Fortran or Java possibilities for Lucas Sequence calculations. So what we are looking for is that the Lucas Sequence predicts some concept of Quantum Mechanics referenced in [1] that can extend the Quantum Field Theory to the Dirac equation, Klein Gordon, Proca, the massive photon and the Theory of Everything in which the dimensions of the Theory of General Relativity are considered by Kaluza-Klein calculations? At first the article is just an explanation of the Lucas Succession calculations in Fortran and the path of Born Jordan Matrix Quantum Mechanics and its eigenvectors, a tentative.