The Schrodinger Equation in Terms of Flux

 In previous notes, it has been argued that the time-independent Schrodinger equation may be written in terms of complex conditional probability, namely:

[Sum over p  p*p/2m fp exp(ipx) ] / W(x) = .5m v(x)v(x)  where v(x) is the classical velocity and W(x) the wavefunction.  ((1))

In this note, we try to understand the Schrodinger equation only in terms of flux momentum flow. We note that:

d/dx density(x) = 2 density(x) u(x) where u(x)=[d/dx W(x)]/ W(x)   ((2))

d/dx u(x) = -m  v(x)v(x) - u(x)u(x)/m  ((3))

d/dx entropy density(x) =  2 u(x) entropy density(x) - 2 density(x) u(x)  ((4)) and

d/dx (.5m v(x)v(x) density(x)) = Force(x) density(x) + .5mv(x)v(x) 2 density(x) u(x)  ((5))

Equations ((2)) and ((4)) are equivalent. If u(x)=0 in ((2)), density should still change due to the second derivative of density (and higher derivatives). In this note, we argue the time-independent Schrodinger equation follows from flux considerations if one writes d/dx density(x) = 2 density(x) u(x) or  u(x)=.5[d/dx density]/density and uses integral dx density(x) u(x)/m u(x)/m= integral dx density(x) v(x)v(x) to show that d/dx d/dx density(x) = 2 density u(x)u(x)/m - 2 density(x) m (v(x)v(x). Here v(x) is the classical velocity given by sqrt[ (2m) (E-V(x)) ].