Probability current proportional to density

[Peeter]

Working the rectangular barrier penetration problem (am working through chapter 11 of the Dover QT book by Bohm) one finds that the current past the barrier is proportional to the current where the proportionality is velocity like:

$$J = \frac{ p \rho }{m}$$

where, the p/m factor has dimensions of velocity:

$$p/m = \sqrt{2E/m}$$

This was under with a steady "stream" of incident wave functions (not a square integrable wave packet).

I find that this proportionality doesn't hold in the barrier region, and was wondering under what circumstances would one generally find the current and the density linearly related like this?

EDIT: I have a guess about this after doing a bit more of the math. J is constant in all three regions (which makes sense given the continuity equation since there is no time dependence in the probability density). Past the barrier we have no interference with flows only coming from the "left". Because of the lack of interference we've also got a constant probability density, so only in this region do we have the velocity-like J and rho linear dependence.

 

[Greg Bernhardt]

@Peeter did you find any more insight on this topic?

 

[Peeter]

@Greg Bernhardt -- It's been so long since I'd attempted that problem, that I'd probably have to attempt it anew to gain any additional insight.

 

[Greg Bernhardt]

I'd probably have to attempt it anew to gain any additional insight.

let's do it! :)