Classical and quantum Hamilton formalisms for the mechanics of the Bernoulli oscillators II : Quantum framework

In a few previous papers, we discussed the fundamentals of the so-called Bernoulli oscillators physics. The Bernoulli oscillators are classical entities whose behavior is influenced by a ''hidden'' degree of freedom, in its turn excited by a quantum vacuum action. The investigation brought us - within assessed approximations and limits (uni-dimensional motion) - to propose a Newtonian motion background below the matter-wave behavior. In the present couple of papers, we give a formal description of the Bernoulli oscillators classical degree of freedom mechanics, by the means of both a classical (in paper I) and a quantum-like (in the present paper II) Hamilton formalisms. The quantum procedure operates on what we call the Bernoulli Hamiltonian functions. There are many different sets of parameters bringing the quantum formalism to the desired fits, so that the most interesting (at the present investigative stage) cases are discussed. We provide finally an ''optimal'' framework, operating on the particle-vacuum primary interaction (a source context to us, called PHME), and we show it able to take out the distant effects of the interaction. This procedure describes our contexts with improved compactness and interpretability. Physical interpretation matching the mentioned formalisms is provided step-by-step.