  The present paper shows that Edward Nelson's stochastic mechanics approach
for quantum mechanics can be derived from the two classical elastically
colliding particles with masses M and m satisfying a collision momentum
preserving equation. The properties of the classical elastic momentum collision
expression determine the full Edward Nelson energy collision energy for both
particles. This classical total energy expression does not require a
statistical expectation since no process was defined for the energy and it
models the main and incident particle velocities perfectly. Quantum mechanics
can be obtained by modelling the incident particle as a non-random potential
using stochastic processes modelling the forward, post-collision and backward
pre-collision velocities of the main particle. This presents the Schroedinger
equation exactly the way that Nelson proposed in 1966 except for the diffusion
constant. In this case the average energy is conserved in time and the forward,
post-collision and backward pre-collision velocities of the system are related
using statistical methods. If the incident particle does not have a potential
the additional constraints for the movement of the incident particle leads to
another Schroedinger equation. Finally, under suitable conditions it will be
shown that the colliding particles satisfy Minkowski metric in special
relativity. This last example shows how gravity can be quantized using details
of this energy expression.
