Klein paradox in the massless case

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SUMMARY

The discussion centers on the Klein paradox in the massless case, particularly at the limit where the energy of the particle equals the potential step height (##E_0 = V_0##). The wave function becomes constant (##k=0##), leading to the Dirac equation yielding trivial results (##0 = 0##) for the spinor components ##\Psi_1## and ##\Psi_2##. The user concludes that, due to the continuity of the transmission coefficient ##T(E)##, the transmission coefficient remains ##T=1## even in this limit, suggesting that an electron must pass through a potential barrier without changing its group velocity.

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Paul159
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I have a question about the Klein paradox in the massless case, for a potential step of height ##V_0## (this is exactly the situation described by Wikipedia). I don't have a problem to understand the "paradox", and I think the Wikipedia's illustration is quite telling.
My question is : what append at the limit case ##E_0 = V_0## ? The "wave function" after the step is constant (##k=0##), and the Dirac equation for the spinor's components ##\Psi_1##, ##\Psi_2## become ##0 = 0##... Thus there are no conditions for the value of those components. If I choose ##\Psi_1 = \Psi_2 = 1## for example, I still get ##T=1##. If I choose ##\Psi_1 = -\Psi_2 = 1##, I get ##R=1##. I "understand" this with the fact that at the node the group velocity is not defined.
So what would really happen in real life ?

400px-Dispersion1.png
 
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Edit : By continuity of ##T(E)## I would say that the good answer is ##T=1##. Also if I end the step potential (I take a potential barrier), the electron coming from the left has to pass the barrier, as it can't change its group velocity. So for me ##T## is always ##1##, even in the pathological case ##E=V_0##.
 

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