Propositional bases for the physics of the Bernoulli oscillators (A theory of the hidden degree of freedom) IV. The matter wave-equation and the Newtonian background

In a few previous papers, we developed a theoretical framework displaying the thermodynamic, mechanical and statistical properties of the Bernoulli oscillators - these last are physical entities to which a classical-like, deterministic behaviour is attributed. We provided expressions for the mass-flow theorem and correlated potentials relevant to an ensemble of these oscillators. In this paper, we compare our framework and results with a quantum mechanical context, represented by the Schrodinger wave-equation in the Madelung formulation. By a requirement of consistency with the quantum equations, we are able to find out an expression of the mechanical energy theorem governing the (single-particle) so-called classical degree of freedom. This expression corresponds to a Newtonian-like equation of motion, which seems to us the good candidate to set a bridge between classical and quantum physics. It is (by proposal) interpreted as the physical background underlying the quantum wave-function formalism. We use a "double solution" conjecture to solve our equation, and show that the classical-like densities generated by the solutions can be summed indeed to the corresponding quantum density. Although we remain sometimes within the boundary of a conjectural framework, and limited to the case of translational motion, the possibility to approach a solution to the old problem of inconsistency between classical and quantum mechanics is conclusively displayed in the paper, and discussed as a proposal.