1. Jun 5, 2009

### JaWiB

I've been doing quite a bit of reading on the Klein Paradox, though I have to admit a lot of the math goes over my head. So I was hoping you guys at PF could clarify a few things and help me check my conceptual understanding so far.

Which seems to partially contradict what I've learned so far; specifically, that the Klein paradox is NOT the fact that the transmission and reflection coefficients don't add up to one. In fact, one source (this preprint) I've found specifically says that this idea came from a mistake in formulating these coefficients and therefore many articles and books are in error in citing negative transmission and greater than unity reflection as THE "Klein Paradox" (the actual paradox is that reflection diminishes as the potential increases, if I understand correctly). Right now, I'm concerned with how one actually comes up with these coefficients.

Could anyone point me to a source that actually carries out the steps of applying the continuity condition on the Dirac equation? Or at least it would be helpful if someone could describe how you go from:
$${\kappa}=\frac{p}{k}\frac{E+m}{E+m-V}$$
to
$${\kappa}=\sqrt{\frac{(V-E+m)(E+m)}{(V-E-m)(E-m)}}$$
(by choosing the appropriate p--see the link to the article I posted)

It seems like it should be obvious (especially since all the sources I've found skip that step) but I don't see how you do it...Plus I'm not even sure what 'k' (not kappa) is in that equation?