Klein Paradox: Clarifying Concepts & Coefficients

[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.

I found this post: https://www.physicsforums.com/showthread.php?t=178587&highlight=klein+paradox

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?

 

[eljose79]

Hello,

Thank you for reaching out to us for clarification on the Klein Paradox. I can understand how the math and terminology can be confusing, so I will try my best to explain it in simpler terms.

First, let's start with the definition of the Klein Paradox. It is a phenomenon in quantum mechanics where a high-energy particle can penetrate a potential barrier without any loss of energy, even though it would require more energy than it has available. This is contradictory to our classical understanding of physics, where a particle would need enough energy to overcome the barrier.

Now, let's address the issue with the transmission and reflection coefficients. The transmission coefficient represents the probability of a particle passing through a potential barrier, while the reflection coefficient represents the probability of the particle being reflected by the barrier. In classical mechanics, these coefficients should add up to one, as the particle can either pass through or be reflected. However, in quantum mechanics, there is also a probability of the particle tunneling through the barrier, which is why the coefficients may not add up to one.

Now, to your question about how to derive the transmission coefficient in the Dirac equation. The equation you mentioned, {\kappa}=\frac{p}{k}\frac{E+m}{E+m-V}, is known as the dispersion relation, where p represents the momentum of the particle, k represents the wave number, E represents the energy of the particle, and m represents its mass. The value of k is related to the wave function of the particle, and it is given by k=\sqrt{\frac{2m(E-V)}{\hbar^2}}. By substituting this value of k into the dispersion relation, you can derive the expression for the transmission coefficient as {\kappa}=\sqrt{\frac{(V-E+m)(E+m)}{(V-E-m)(E-m)}}.

I hope this explanation helps clarify your understanding of the Klein Paradox and how to derive the transmission coefficient in the Dirac equation. As for sources, I would recommend looking into textbooks on quantum mechanics or specifically on the Dirac equation. You can also find some helpful resources and tutorials online that can guide you through the derivation process.

Best of luck in your studies! Please reach out to us if you have any further questions or need clarification on any other concepts related to the Klein Paradox.

 

[Entropia]

First of all, it's great that you're doing your own research and seeking clarification on the Klein Paradox. it's important to constantly question and seek understanding.

To clarify, the Klein Paradox is not simply the fact that the transmission and reflection coefficients don't add up to one. As you mentioned, this idea was a result of a mistake in formulating the coefficients and is not the actual paradox. The paradox stems from the fact that in the presence of a high potential barrier, there is a non-zero probability for a particle to tunnel through the barrier, even when its energy is less than the barrier's height. This goes against classical physics, where particles would be completely reflected by the barrier.

Now, to answer your question about how to come up with the transmission and reflection coefficients, it involves applying the continuity condition on the Dirac equation. This means that the wavefunction and its derivative must be continuous at the boundary of the potential barrier. The equation you have provided, with {\kappa} and k, is related to the momentum of the particle and the potential energy. The exact derivation of the coefficients can be found in many textbooks on quantum mechanics or particle physics, and I would recommend consulting those for a detailed explanation.

In summary, the Klein Paradox is not simply the discrepancy in the transmission and reflection coefficients, but rather the counterintuitive phenomenon of particles tunneling through high potential barriers. The coefficients are derived by applying the continuity condition on the Dirac equation, and the exact steps can be found in textbooks on the subject. Keep up the good work in seeking understanding and questioning concepts in science.