Quantum and Classical Mechanics and Two Delivery Schemes for Force Part II

  Newtonian mechanics deals with two types of force delivery schemes, one linked to the variable x and the other to t. They may be written as:  Integral Fdt (impulse) and Integral Fdx (work) and appear to be symmetric in the two variables. The problem is that expressed in terms of dp one sees an immediate difference linked to discrete versus continuous change. In particular:  dp= Fdt ((1a)) and  vdp = Fdx ((1a)).  ((1a)) allows for a discrete change in p in a tiny interval of time. “X” does not come into play at all in the equation and one might think of the impulse occurring at a point x. Due to the discrete nature of integral dp changes, one may think in terms of probabilities because a given pA may have been obtained from discrete impulses:  p1→p2→p3  or p4+p5. etc. The final p state is the same and so not only does there exist probability but also the notion of product probabilities such that exp( pi b) exp( pj b) = exp( (pi+pj) b) where b is a constant. Given that exp(pb) increases exponentially, one might rather use the periodic probability exp(i p b). One may note how impulse type hits and their discrete momentum results lead to stochasticity in a Maxwell-Boltzmann gas. 

   The form  vdp = Fdx is very different in that it does not allow discrete jumps of p because v=dx/dt. If one assumes a very large F at x1 and dx–0 such that the product is not 0, one might consider dp changing discretely, but what is then used for v? Instead one considers a range of x with a corresponding  range in t such that x(t) in order to compute v. Thus this is a deterministic situation because x and t cannot be decoupled. It leads to the notion of  a kinetic energy pp/2m associated with p. This kinetic energy is calculated by considering p change from 0 to p.

   An interesting feature arises because impulse = Integral Fdt may be thought of as discrete allowing p1 to jump to p2. Nevertheless, the deterministic approach states there is a kinetic energy associated with p1 and p2, namely p1p1/2m, p2p2/2m, even though no deterministic path in time and space was taken. The impulse was said to occur at x in a very tiny time range. Thus there seems to be a contradiction. pp/2m requires a range of x and t with x(t). The impulse only occurs in time, with x decoupled but there must be an associated x range which is then not deterministic i.e. a decoupled x range. This suggests a characteristic length associated with a momentum p in which one does not follow time deterministically. If one does not follow x in t in this special length, then one has a probability distribution. A similar argument occurs for time. Consider Fdx with time absent. If dx becomes very tiny and F(x) very large , one may think that work is done at an instant in time, but there must be a time range for pp/2m. Thus there is a special time period not associated with x in which one has probabilistic behaviour i.e. not x(t).

     One may note from the notion of impulse that a given p may be obtained through different series of smaller impulses, e.g.  {p1,p2,p3} and {p4,p5}. These are associated with kinetic energy series {p1p1/2m, p2p2/2m,p3p3/2m} and {p4p4/2m, p5p5/2m}. The final result is the same. Thus the discrete nature of changes leads to a probability scheme such that P(p1)P(p2) = P(p1+p2) and P(e1)P(e2)=P(e1+e2). This we argue leads to:  P(p) = exp(i p b1) and P(e) = exp(i e b2) where b1 and b2 at first seem to be constant. We noted, however, that an impulse implies a probabilistic range in x and Fdx→finite when dx→0 a probabilistic range in t. Thus b1 may be replaced with x and b2 with t. At any given point in time or space, x and t are constants i.e. b1 and b2.

  Next, we consider the situation of special relativity i.e. a Lorentz boost. We consider an initial state of energy= mo, x=0, t=to and view it from a frame moving at a constant velocity v. Although one may use infinitesimal velocities, a main point we wish to make is that special relativity allows for the use of finite velocities which give the sense of discrete changes like in the impulse case.This should again be associated with probabilities. Special relativity yields the Lorentz invariant  A = -Et+px  where v=x/t.  Nevertheless, x and t are written in the invariant as if they were independent. It may be noted that dA=0 if t changes on its own as hbar/E and x on its own as hbar/p. Thus one may obtain specific x and t probability ranges.This is again consistent with the probabilistic picture or exp(-iEt) and exp(-ipx). 

    It may be noted that even with the probabilistic picture, a particle with constant p moves determinisitcally on average as x= p/m t. If x = hbar/p, then 1/t is not hbar/E and if t=hbar/E then x is not hbar/p. Thus the values  hbar/E and hbar/p seem not to be linked with the current deterministic motion, but rather with the creation of p in the first place. E=pp/2m and this is obtained by considering p changing from 0 to p. Thus the “frequency” E seems to capture the information of the creation of p and we argue that the wavelength captures the creation of x as a probability distribution in x which one would usually associate with force/acceleration.