Average Energy as a Deterministic Attribute Due to Statistics of exp(ipx)

A quantum particle (or photon) is described by the two dimensional probabilistic exp(-iEt+ipx), with t and x being treated as independent. exp(-iEt) suggests a stochasticity in time associated with energy E, but none in space. Similarly, there is a stochasticity in x associated with p but none in time. For -Et+px to be invariant as seen in different frames moving with constant average velocities, -EE+pp must also be an invariant i.e. equal momo (c=1). This leads to a contradiction. exp(ipx) suggests a stochasticity in x for a given p, but -EE+pp=momom converts a given p into an E, so there exists stochasticity in x for E as well (which contracts the notion that exp(-iEt) is only stochastic in time). The link between E and p, however, which breaks x-t independence is the notion of average velocity v=x/t, which is physical (as is exp(ipx)). How does one deal with this contradiction?

    We suggest that p and E are associated with different physical interaction processes. P is related to impulse hits, while E may be associated with loss of energy over a distance if a potential V(x) exists. Even though exp(ipx) suggests stochasticity in x, at the same time one considers fixed objects (targets, slits) and even potentials V(x) at well-defined deterministic x points. This seems to mean that if one wishes to make energy deterministic at each x, one must create an average of exp(ipx)s. 

  We argue that there seem to be two important ideas associated with exp(ipx). First, it is a two-dimensional probability linked to a flux because it is due to p which is due to a v. Probability flux does not get created or destroyed (unless there are sinks or sources). This leads to issues when dealing with a bound state which should be localized. A flux probability is not localized, nor is an average in general (except in the case of an infinite potential). A bound state problem is really a flux problem consisting of a particle trapped in a potential together with it tunneling out or in i.e. one has a dynamic equilibrium picture somewhat like evaporation in equilibrium.  (To see the clear difference between a state based probability and a flux-based one, consider the problem of a photon moving in the x direction and reflecting/refracting from an n1-n2 (index of refraction) y-plane at x=0. State probabilities (P(reflect), P(refract)) are not sufficient to solve the problem, but a flux-based one exp(ipx) which allows a single particle to appear as if in a steady stream scenario is.)

   Secondly, unlike evaporation, the guiding statistical condition for exp(ipx) is that its average makes E deterministic in x space. Thus E is like T (temperature) in that it is a statistical parameter, but it is not based on maximization of entropy (equilibration), but rather in creating E as an average which is then treated as deterministic (ignoring in a sense exp(-iEt) value characterizing an equilibrium system (not based on equilibration). This deterministic E system is linked with discrete values (due to the exp(ipx) equilibrium), but given that E is treated as deterministic, a stochastic process based on maximization of entropy is introduced to describe the absorption and emission of photons (which allow for jumps from one average-deterministic En to another).

   We argue that the flux-like probability exp(ipx) exists in order to deal with sharp interfaces (e.g. n1-n2 index of refraction) at a deterministic x=0 which probabilistically leads to different p values. It uses the reverse approach i.e. a deterministic p and a stochastic x. In such a way, one may use continuity (and first derivative continuity) at the sharp x with flux-probabilities containing the various allowed p values (due to the impulses). The continuity follows from no sinks or sources existing for the flux-like probability.