Quantum Mechanics and Pressure Balance in the Presence of Flux and Momentum Discontinuity

One may consider constant fluxes which exist for all x points in space. These are associated with a constant velocity and a momentum. Consider next a discontinuity which does not change energy. For example, one may have energy E, momentum p and a constant potential V1 which suddenly changes to V2 at x=xo with E remaining constant. This leads to a momentum discontinuity as well as a flux discontinuity, but no work is done. As a result, there exists a pressure balance (related to an incident flux, a reflected one and the new p2 flux for x>xo), i.e. there is no discontinuity in pressure. A single constant flux also has no discontinuity in pressure either. We suggest there should exist a continuous function of momentum and x which allows for/ensures this pressure continuity. The momentum discontinuity may be at any xo, but the pressure balance is the same.As a result, one would expect this function and its first derivative to be continuous at xo, but a combination of the two should eliminate the function (of p and xo) leaving only a pressure balance equation. In other words, two equations need to combine to create a single classical pressure balance equation, suggesting that there may exist a square root flux type probability function.

  We suggest that pressure balance at momentum discontinuity points (with E remaining constant) may be the only consideration necessary to ultimately obtain the quantum free particle exp(ipx). An advantage of this pressure-only viewpoint is that it accounts for weighted averages of exp(ipx) having the same form, namely exp(ipave(rms) x) and exp(-ipave(rms) x) in a potential with infinite walls. Pressure considerations may be applied to a single particle exp(ipx) or an ensemble, yielding the same form if the same conditions apply. We argue that both a square well potential and infinite well potential are examples of momentum discontinuities, except in this case one has discontinuities in ensemble  values (i.e. average values).