Information and Wavelength In Newtonian Mechanics?

  In (1) a particle with rest mass and momentum magnitude p1 moves from a region with a constant potential V1 into a region with a constant potential V2 with the interface being along the x axis. Overall energy is conserved and it is noted that conservation of the x-component is analogous to Snell’s law. ((1) goes on to use a variational approach to argue conservation.)

    In this note we begin with Newton’s idea that momentum changes in the direction of a force and that this force is -dV/dx where V(x) is a conservative potential. For constant V then, forces only arise from discontinuities in potential. We note that for a two dimensional problem (x,y) choosing a starting point of (x=0, y=Y) and changing the incident angle (measured from they axis, leads to different x values. Thus x is a piece of information in the two dimensional problem. In one dimension (along the x axis), however, momentum is conserved because there is no discontinuity in potential. In this one dimension there is x invariance and x should not be a piece of information. The question becomes: Is there a function in two dimensions that when applied to the two regions of this problem leads to x information disappearing? We consider an ending point (L-x, -Y) where L and Y are constant and note immediately that:  x+ L-x=L i.e. a constant so that x disappears. Thus if x is multiplied by a constant (namely the x component of p) then the function A1=(px) x + (py)y meets the requirement i.e. A1+A2 loses all information about x because of the conservation of (px). As a result one sees the association of p and x which in Newton’s theory would normally not be combined. Furthermore if one uses the constraint that L=1/(px) (and drops the y=Y,-Y requirement), then (px)L = 1 which holds for all px values. This is linked to the idea of having one constant (namely 1 here) representing all constant motion as discussed in (2). In this way one may see a characteristic length proportional to 1/px emerge so that (px)(1/px) is the same for all constant motion. This characteristic length is the wavelength which appears in experiments.

   A second issue we wish to consider is the quadratic form of pp in both nonrelativistic E= pp/2m + V(x) and relativistic (E-V)(E-V) = pp + momo  (c=1)  energy conservation equations. We use the refraction type problem (two constant potentials) together with conservation of (px) = p sin(theta). We use Newton’s idea that force changes momentum in the direction of -dV/dx, (py) and hence p magnitude must change, but (px) must be a constant. Thus one may write:  p1p1 - p2p2 = (p1yp1y-p2yp2y) = C1 ((A)). We have used p1x=p2x, but ((A)) leads again to this result so the general result   p1p1=C2 should hold. A similar argument holds relativistically. Thus without integrating one is led to the idea of a quadratic p (i.e. p magnitude p magnitude) as part of a constant equation (becuase the potential is constant). This we argue is conservation of energy.