Statistical View of Quantum Mechanics with Time Included

In a previous note (1), we tried to present a statistical view of quantum mechanics for a bound state. Time was ignored and it was suggested that at a point x, V(x) and kinetic energy are both averages which add to a constant E. It was also suggested that V(x)=Sum over k Vk f(k,x) and KE(x)= [Sum over p p*p/2m f(p,x) a(p)] / W(x) where W(x)=Sum over p a(p) f(p,x). We tried to argue that by using conservation of momentum and introducing statistical uncertainty in x, one may obtain f(p,x)=exp(ipx). In other words, if a particle is in a momentum state p1 at x at an instant in time t1, it may be in p2 at t2, due to the action of the potential . Thus, to measure energy takes a certain interval of time. This seems to suggest an uncertainty in measuring energy and time together. Furthermore, the interaction of f(p,x) with Vk f(k,x) leads to the particle jumping from one state to another, so as energy is being measured, the state represented by the normalization constant W(x) may be evolving i.e. W(x,t). In this note, we try to consider aspects of time in this statistical picture, which are present even in the bound state case, as has been noted in previous notes, which started with exp(ipx). In this note, we try to argue that quantum mechanics seems to follow directly from statistical arguments, without the need for exp(ipx) or operators as a starting point