Quantum Bound State Phase exp(i dx < p at x>) and Hydrodynamics

In a previous note (1), we argued that W(x+dx)= W(x) exp(i dx <p at x>) where <p at x>= - i d/dx lnW mimicking the behaviour of exp(ipx) for a free quantum particle. In this note, we consider that for the conditional probability P(p/x)= a(p) exp(ipx)/W(x), a shift from x to x+dx leads to: P(p/ x+dx) = P(p/x) exp(i dx (p-<p at x>) ) = P(p/x)(1+ idx (p-<p>)). We note that in shifting from x to x+dx the overall sum of probability should not change, so P(p / x+dx)= P(p/x) + P(p/x) g(p,x). If one assumes that g(p,x) is a function of p alone,  the simplest solution is p-<p>.  If one considers  Sum over p pp/2m P(p/x+dx) = Sum over p pp/2m P(p/x) = delta KE ave then this should be equal to Force(x) dx. 

   We also consider classical hydrodynamical equations and compare these with modified equations applying to the bound state quantum situation. The two sets of equations are very similar with density in the classical case being replaced with W(x)exp(-iEt), the wavefunction, and E energy or V(x) (potential) terms replaced with 2E or 2V (i.e. also 2Force). In a previous note, we had considered hydrodynamical equations with the restriction 2m=1 which is too strong. Here any m (mass) may be used.