Quantum Mechanics and Writing the LHS of dp/dt = Force in terms of a Gradient? Part 2

In Part 1, we tried to obtain free particle quantum mechanics by writing the LHS side of dp/dt = F as a gradient because the RHS F = - grad V(x) may be written in this manner for a conservative force. Here we argue that just as V(x) has physical meaning, p = grad partial (-Et+px) (found in (1)) should represent the physical potential created by a free particle. That does not change the fact that the particle center-of-mass moves as x=vt, but the interaction created by the particle should be described in terms of a potential and this was found to be -Et+px. From the point of view of a force system (e.g. a 2-slit system) interacting with the potential, one needs a potential created by the free particle. The potential -Et+px shows that an interaction at t,x has the same potential value as one at t+hbar/E and x+hbar/p. Given that this potential follows from dp/dt=Force, we suggest it should be taken seriously and that it indicates there is the possibility of interaction at two different x points x and x+hbar/p in a stochastic manner. In other words, a single free particle could interact with one slit or the other of a 2-slit system if the slit separation is about hbar/p. This still leaves two questions. First, is such a two fold interaction deterministic or probabilistic. We suggest probabilistic because there are ranges for x and t. Second, why does the usual Lagrangian method of finding a stationary solution from d Integral dt L(x, dx/dt, t) = 0 (leading to d/dt dL/dv partial - dL/dx partial = 0) not show this range in delta x? We suggest that writing p = grad partial (-Et+px) (so the potential is Lorentz invariant) removes the dynamics of the particle from the interacting force system (2-slits). The 2-slit is interacting with a free particle potential (not with its dynamic motion. The two need not be the same. We suggest that there are two theories for describing a particle interaction, the single particle potential theory and Lagrangian formalism. We thus suggest that they must be linked and try to show how they compare and how they differ in a systematic manner.