Time-Independent Schrodinger Equation as Balanced Pressure-Velocity Flow Equation?

In a previous note (1), it was shown that for a ground state oscillator wavefunction, one could analyse the problem in momentum space as a free gas of momentum particles and obtain exp(-p*p/2mT) as fp*fp, fp being the momentum wavefunction. A free energy equal to zero was obtained, with kinetic energy being the energy term in E-TS.  This suggests quantum particles scatter (with conservation of momentum) like classical point particles in momentum space, but differ from classical point particles in position space. In particular, one takes a Fourier transform of fp as it seems that each momentum wave carries an exp(ipx) function which is a sort of “baggage” term, possibly due to zitterbewegung. This seems to explain why a free gas in momentum space for the ground state oscillator yields a Gaussian density with W(x)=exp(-ax*x) (W(x)=wavefunction). An oscillator potential is then needed to keep this density with its momentum space pressure in balance. Thus, quantum mechanics and classical statistical mechanics become very similar for this scenario. This begs the question: What about other potentials or oscillator states higher than the ground state? In such a case, it still seems that one may consider “particles colliding in momentum space” with each carrying an exp(ipx) “baggage” term, but one does not have a “classical statistical mechanical” equilibrium scenario any longer. Rather, it seems that one has a net force created by -dV/dx (V(x)=potential) and a difference in pressure d/dx [Average(p*p)d(x)] where d(x) is density. There also seems to be an average velocity  (1/W)dW/dx. In momentum space such a velocity would not exist for f(p)=f(-p), but because of the baggage of exp(ipx) it is present i.e. p exp(ipx) - p exp(-ipx). Newton’s law, with force acting on this velocity appears in the equation d/dx [Average(p*p)d(x)]=F(x) d(x).

This same equation is equivalent to the gradient of the time-independent Schrodinger equation.       

 For quantum mechanics, one has one particle rather than many as in classical statistical mechanics. Thus, it seems that this “Newton’s law” balance is based on a long term time average. It seems, however, that there should be a period T during which the average can establish itself. After this, one would have periodic time motion where the system keeps scattering back into itself in this characteristic time. This period can be deduced, it seems, from the time-independent Schrodinger equation and is found to be E, the energy.