Newton's First and Second Laws Leading to Different Spatial Probabilities One Being Quantum Mechanical? Part II

Addendum Nov. 27, 2022. In Newtonian mechanics, one may slice the x axis into equally spaced dx with a particle being in one dx at time t1 and in the next at t2. In this note we suggest describing directional constant motion with a probability density P(p,x). If the particle moves out of x we suggest that;. P(p,x) dP(p,x)/dx=0  or P(p,x) must be at least two dimensional i.e. P(p,x) = F(p,x) + iG(p,x) such that  F dF/dx + G dG/dx =0. This is equivalent to:   FF + GG = constant for any x which shows all x points being the same even though P(p,x) is a function of x which distinguishes direction. Finally, with P(p,x)P(-p,x) = constant because this represents no motion and so no distinguishing density is needed one has P(p,x) = exp(ipx).

In Part I we argued that directional motion should be associated with a different spatial probability densities. dtime(x)= dx/velocity(x), however, taken as proportional to a density in the literature (1) does not distinguish. We suggested that Newton’s first law is equivalent to Fermat’s least time principle which for reflection from a fixed mirror is equivalent to d/dx (px) x = (px). (px)x, however, may be written as -i ln[exp(i (px) x) ] which has the form of constant multiplied by ln probability in information theory. Thus we suggested in Part I that exp(ipx) is a kind of probability density which distinguishes direction in motion. We also noted that exp(-ipx) exp(ipx) = 1 which is linked to the dt(x) density for constant motion and that  Integral exp(-i p1 x) exp(ipx) dx = constant delta (p1, p). 

    In this note we wish to examine in more detail the notion of x, t and wavelength. exp(ipx) depends on x and p=mo dx/dt or (p=mo v / sqrt(1-vv/cc) with v=dx/dt). X and t are continuous variables linked to the idea of invariance in space for constant motion. exp(ipx) and exp(-iEt) on the other hand define built-in particle space and time intervals for a given p,E. Thus exp(ipx) defines constant motion in space without the notion of time (i.e. t and x are independent within a built-in ruler gradation proportional to 1/p (two humps-one positive one negative). This leads to a contradiction if the particle is tiny compared to 1/p. 

   On the one hand the particle has smooth motion v=x/t, but on the other a “created” particle  length unit must be defined suggesting a probability density over the length proportional to 1/p. To allow for the two, one must assume a constant average motion over the length 1/p as measured by an external ruler and clock and at the same time abandon the notion of following the particle in x,t (infinite resolution) within the length proportional to 1/p even though its probability distribution exp(ipx) is well defined at each x. 

  Thus the identity of the particle is defined by the probability exp(ipx) within 1/p regions even though a direct hit with another particle within this region involves p.  Furthermore given that the spatial probability defines the particle it is not enough to state that for two particles one has p1 and p2, the spatial probabilities must distinguish by being orthogonal i.e. having a 0 dot product: Integral dx exp(-i p1 x) exp(ip2 x). We argue in the note that not only is particle identity described by exp(ipx) and exp(-iEt), but also conservation of energy and momentum in various problems.

    In Part I we suggested considering the product of P(p,x) and P(-p,x) in order to obtain a constant value linked to spatial variance throughout x even though exp(ipx) and exp(-ipx) are not spatially invariant. We first note that exp(ipx) and exp(-ipx) are the time reversal of each other and then consider exp(ipx) AND exp(-ipx) together within a little region of the order of 1/p. exp(ipx)exp(-ipx)= 1, but each moves with its own density profile expx(ipx) and exp(-ipx). Thus constant density ignoring the direction of motion seems to be identical to the product of the time reversed density states.

   Even though constant motion is only a translation when viewed on average, we suggest that directional motion must be linked with skewed density and be based on internal distances and clocks. This contradicts the idea of a smooth translation and leads to an unusual probability description within hbrar/p wavelength units. The catch seems to be, however, that one needs to introduce two dimensions in order to demonstrate the skewness linked with directional motion.

    By creating a spatial wavelength, the region within is one in which one may not follow the particle in x and t i.e. x and t are independent as in the classical action: A = -Et+px. dA/dx partial = p, but one cannot write this unless x and t are independent and only represent constant v on average i.e. x/t. Finally the wavelength and interval in time, which are physical, are directly linked to ensuring conservation of energy and momentum in the direction of an invariant variable. We argue this by consider a two dimensional reflection problem i.e. Fermat’s principle.