Potential-Force and Two Component Quantum Wave exp(ipx)

    In quantum mechanics, a free particle is represented by exp(ipx). It has often been noted that exp(-ipx) exp(ipx) =1 yields equal probability for any point x. In classical mechanics, one might argue that a constant real number suffices. The form exp(ipx) is also identical to the classical electromagnetic expression associated with light (i.e. a photon). In this note, we wish to begin with the idea of an ensemble of momenta at x using a real probability f(p,x) and create a conservation of energy equation i.e. KEave(x) + V(x) = E. This equation may be differentiated to yield a force equation with d f(p,x)/dx. If one argues that <p ave at x> should appear in such an equation, one has a link between d f(p,x)/dx and <p ave at x>. One finds that f(p,x), which we assume to be the same for p and -p so that pf(p,x)-pf(p,x)=0 as in classical statistical mechanics, may be generalized to f(p,x) + i g(p,x) with g(p,x) linked to d f(p,x)/dx and <p ave at x>. 

   In other words, quantum mechanics includes two components, one associated with V(x) and <pp/2m at x> and another with F(x) and <p ave at x>. The Bohm Aharonov effect seems to justify such an approach because one may have V(x) non-zero and F(x) zero and still have physical effects, thus a potential V(x) and F(x) may both have separate physical consequences. Using cos(px) and isin(px) seems to track these two separate consequences through the separate consequences of <p at x> and <pp/2m at x>. For a classical particle, p becomes pp by squaring so the two do not need to be treated with two separate “dimensions’ i.e. cos(px) and isin(px).

  In this note, we try to obtain the form exp(ipx) from the more general exp( i Theta(x,p)) without resorting to the form exp(ipx) of a  photon in classical electromagnetism and also consider an interpretation.