Importance of Newton's Action-Reaction for Free Particle Quantum Mechanics

    In a number of previous notes, we argued that free particle quantum mechanics, i.e. exp(-iEt+ipx), follows from the Lorentz invariant A = -Et+px. For t=0,x=0, A=0, but we argued that there is another solution, dx = hbar/p and dt=hbar/E  (we use intervals because x=0, t=0 is a trajectory point). We then noted that this solution is linked with the probability exp(-iEt+ipx) which is Lorentz invariant and creates the above intervals in t and x. Finally, we remarked on the fact that exp(-iEt) conserves energy and exp(ipx), momentum, when one uses these for 2-body collisions, i.e. AND situations.

    Here we point out that there are numerous dt= f(p,E) and dx=g(p,E)  (f, g are arbitrary functions) which yield A=0. For example, dt=p and dx=E is an example. Only dx=hbar/p and dt =hbar/E, however, lead to a probability form which conserves momentum and energy and this is something which we did not stress clearly in previous notes. Thus, there seem to be a number of issues involved in obtaining a free particle probability exp(-iEt+ipx) and these are independent requirements, which then lead to a particular solution. We argue that these requirements are:

  ((1a)) One wishes to preserve Lorentz invariance

  ((1b))  One desires discrete dx, dt intervals instead of Newton’s intervals which tend to 0.

  ((1c))  These intervals need to be linked with a particle related probability which defines them. 

             This probability is critical physically because if the theory is to define photons and   

              particles equally well ( as special relativity ((1a)) applies to both), one must account for     

              the probabilistic effects of 2-slit interference and 1-D reflection-refraction at an n1-n2 index of refraction junction.

   ((1d))  One must have conservation of momentum and energy.

We argue that it is the combination of these requirements that ultimately leads to the free particle probability exp(-iEt+ipx).