Consequences of a Particle at X and a Force Acting Over dx and Quantum Mechanics

Classical Newtonian mechanics separates a particle (effect) from a force (cause). A particle has properties at a given x (and time t), but if it interacts with a force, this interaction occurs over dt and dx. Given that many forces do not change in time, we argue that a notion of a force is inseparable from the idea of dx. If one gives the value of a force at x, this value acts within a dx, which in Newtonian mechanics tends to 0. 

     We have argued before that a particle with momentum represents a force or impulse itself, but at the same time may receive impulse hits from other sources (e.g. a Newtonian force -dV/dx). If a particle represents a force(impulse) it should be inseparable from the notion of dx, even if it has a constant p (momentum) and is traveling through space. Thus the idea of a momentum defining an internal ruler gradation through a probability distribution in x which we have presented in previous notes is compatible with the idea of describing a particle as a force. If one only considers a particle as an effect, there would be no need to worry about dx until a force(x) appears. 

    An internal dx is not defined by an external ruler and does not need to tend to 0 as in Newtonian mechanics. Rather it is statistical in nature as p is a fixed impulse, but the probability to be at different x points within dx (called a wavelength) differ. In (1) we compared a classical Maxwell-Boltzmann gas to a quantum free particle. We argued that d/d density Pressure = RT = average pressure of a particle. We suggested this is analogous to the quantum free particle situation:   (-id/dx probability(p,x) ) / probability(p,x) = p   where probability(p,x) is unnormalized. In other words a change in a probability yields a deterministic value p. 

     The notion of a deterministic value p associated with a probability distribution in what would be another deterministic variable x in Newtonian mechanics (except now one has a dx distribution because the particle is a force/impulse) has consequences. If one wishes to have a particle sit at x=0, say in an oscillator system, one needs to use a wavelength which is fairly small and centered at x=0. The wavelength, however, is associated with momentum and even being small spreads out from x=0 meaning there is potential energy .5kxx as well. Thus one may use an ensemble of forces/impulses  Sum over p a(p)exp(ipx) to create a “dx” about x=0, but at the same time this ensemble represents an average particle (defined at x with prms(x)) which must satisfy KE(x) + V(x) = En i.e. the classical conservation expression. In other words, one sees again that average statistical quantities may behave as deterministic ones.