Speculation on Internal (Momentum) and External (-dV/dx) Forces in Quantum Mechanics

  Newton’s second law dp/dt = Force seems to be written in terms of a cause F(x) and effect p(x), because momentum is written in terms of a change i.e. d/dt. His third law, however,describes two causes i.e equal and opposite causes (forces) leading to two effects. We argue that a constant momentum, the effect of a force, may also act as a “force” if divided by a constant time i.e.  change in p / dt(constant) = Force. A conservative external force may be written in terms of the derivative of a potential i.e. -dV/dx, but we suggest that a quantum free particle is linked with two distributions cos(px), sin(px) which describe an internal force/acceleration for even a constant p. The two combine in two dimensions i.e. exp(ipx) to create a type of probability which has a constant modulus i..e treats each x point equally. Each x point is treated equally because there is no external force -dV/dx, but we argue that there is still an internal one linked with probability and spatial distributions i.e cos(px),sin(px). If an impulse is applied to a quantum free particle to change it from p1 to p2, the change in energy is completely described in terms of momentum i..e p2p2/2m - p1p2/2m. There is no need to consider space, but at the same time there is a change in the two distributions cos(px) and sin(px) due to the change in p. 

    Thus a constant p is linked with two spatial distributions which have physical consequences i.e. two slit interference etc. There is an internal distribution in space. If a quantum particle is placed in a potential V(x), there is now both an external potential, but also an internal one associated with each exp(ipx) if one assumes an ensemble of exp(ipx)’s is created i.e. W(x)=wavefunction = Sum over p a(p)exp(ipx). The point we make is that both the external and internal potentials are linked with x distributions, thus applying d/dx effects both, but Newton’s conservation of energy KE(x) + V(x) = E deals with changes in kinetic energy only linked to the external potential V(x). Formally, one may write:   -1/2m d/dx dW/dx / W + V(x) + E, but we argue that external and internal potentials “interact” and we try to describe the situation considering V(x)W(x) to demonstrate the interplay of internal and external forces from the point of view of the internal force (probability) exp(ipx) and statistical mechanical loss of information i.e. equilibrium.