Bound Quantum Mechanics, Classical Mechanics and Flow/Flux Scheme

 In a previous note (1), we argued that both free particle quantum and classical mechanics follow from an energy flow/flux scheme based on:

d/dx (partial) A(x,t) = p  ((1a))   and d/dt (partial) A(x,t) = -E  ((1b))

A(x,t) turns out to be the classical action tL(v) (L=Lagrangian)  with x/t=v and the above hold for both the relativistic and nonrelativistic cases.

We further argued that ((1a)) and ((1b)) lead to two sets of solutions, one classical and the other quantum. The classical solution assumes dt→0 and dx→0 i.e. flows may be followed at each point in space (and time).  In (2) we argued that ((1a)) and ((1b)) are consistent with  p=dL/dv (partial) and H=E= pv - L. From these together with p=Ev one may find: E=mo/sqrt(1-vv)  (c=1). For the nonrelativistic case, E= mocc + .5mvv. In this classical analysis one may resolve each t and x, thus if one has a function V(x) (potential) representing energy, the nonrelativistic result is:    (E-V(x)) = .5mvv.

     The quantum solution to ((1a)) and ((1b)) is very different. It establishes eigenvalue equations i.e.  id/dt (partial) exp(iA) = E exp(iA) and -id/dx (partial) exp(iA) = p exp(iA) for the free particle. The importance of this is that new physical mechanisms seem to come into play. A= -Et+px so exp(-iEt+ipx) suggests time and spatial regions proportional to 1/E and 1/p in which there is some ambiguity i.e. there are two dimensions and there seems to be some kind of probability for the particle to be in one dimension or the other at x (and similar results for t) i.e. cos(px) and sin(px). As a result, one may not have dx→0 and dt→0. Furthermore, it is argued that a two dimensional associated probability scheme must accompany energy flow. At first this may seem strange, but if one considers a photon which has an energy flow Ev, one has electric and magnetic fields perpendicular to each other and perpendicular to the motion which undergo periodic motion linked to cos(wt) and sin(wt). Thus, the quantum particle periodic probability mimics an already known result from Maxwell’s equations. In fact, we have argued that for fermions there is additional angular motion i.e. spin linked with the recognition of 3-space. Thus we argue that quantum mechanics is involved with periodic type probability motions which are needed to resolve dx and dt  (linked to exp(-iEt+ipx) and periodic probability motion (spin)  and are also needed for fermions to resolve 3-space as two p vectors may have point in different directions (i.e. spin).

  Quantum mechanics therefore seems to be a very different theory from classical mechanics, but both appear to be based on energy flow/flux. As a result, the average of a quantum bound state should be linked to a classical system we argue in this note as one only deals with energy flow/flux.