Time Base Force (Impulse) Versus Space Based Force (-dV/dx) in Classical and Quantum Mechanics Part II

In a previous note, we argued that there are two scenarios in classical mechanics, the idea of an impulse dp=Integral F dt which imparts the same dp to any p and a F=-dV/dx with pdp = Integral F dx which imparts a momentum based proportional to 1/p. Generally, F=-dV/dx is used in a Newtonian bound problem (e.g. particle on a spring etc), but it is the impulse scenario which yields conservation of momentum. Given that V(x) is used in classical physics, it was also applied to small regions i.e. those of an electron in an atom etc., but non-classical results emerged experimentally leading to discrepancies. In this note like in Part I, we try to retain V(x) on average, but argue that interactions of a single bound particle with a potential should be of the form of statistical impulses. Force=-dV/dx changes p to p1 in a 1/p dependent manner over dx (i.e. at the point x) while an impulse delivers a dp independent of p. Thus, it seems a kind of averaging is needed for impulse p values to create the appearance of an average change in kinetic energy or acceleration. The idea of -d/x Function(p,x) is retained for an impulse (as a force), so Function(p,x) is a “piece” of the potential which should be invariant over space and -d/dx Function should be proportional to p i.e. the hit given. It cannot be a constant, but the next best thing is to have exp(ipx). Thus, considerations of impulses, invariant in space, may create the appearance of a V(x). We argue that exp(ipx) is of the nature of a “probability” and that it has consequences on the “probability” of the single particle with which it reacts. In fact, it requires the single particle to also have a “probability” of exp(ipx). This leads to a probability which is “spread” out over a wavelength, much like a wave in classical physics. Thus, there is the idea of discerning a p at point x as in classical physics versus discerning it over a range of space as in quantum mechanics. We consider some of these ideas in this note