Talk:Klein paradox/Archive 1

Archive 1

Massless particles

Please explain, why the wavefunction $\psi$ is now not a bispinor! Where are the coupled differential equations from the Dirac-equation? 141.89.36.156 (talk) 13:59, 2 July 2014 (UTC)

Give Credit Where Credit Is Due

While the authors of the recent publications did explain what the significance of the spurious wave is, they cannot possibly be said to have "resolved" this paradox. To resolve a paradox requires that the paradox should still be confusing at the time of publication.

Historically, the Klein paradox was important in demonstrating that a single-particle Dirac equation is inconsistent, and antiparticles are really required. This is its main historical purpose. It was resolved by Dirac, Klein, Heisenberg and whoever else did second quantization way back when. The recent publication is an analysis of the significance of Klein's solution. While it is interesting, and worth citing, it is absurd to say that the paper "resolved" the paradox just because the word "resolution" appears in the title, with the meaning in "the resolution of my monitor is 760 by 1200". Likebox 02:53, 30 September 2007 (UTC)

Use motion along z instead of x

I like the simple model with a massless two-component fermion. However, instead of considering motion along the x-axis, the argument would become even clearer when using the motion along the z-axis as the Pauli Matrix sigma_z is diagonal, in contrast to sigma_x. 195.126.85.141 (talk) 15:25, 11 October 2010 (UTC)

Probability current

$J_{i}=\psi _{i}^{\dagger }\sigma _{x}\psi _{i}^{\dagger },i=1,2\,$

Shouldn't it rather be

$J_{i}=\psi _{i}^{\dagger }\sigma _{x}\psi _{i},i=1,2\,$

? Sigi E (talk) 09:31, 17 January 2011 (UTC)

Yup, fixed - thank you. Feel free to be bold in fixing things yourself. - 2/0 (cont.) 21:14, 17 January 2011 (UTC)

Region II solution

For the region II solution as given one has $k<0$.

The region II solution for $k>0$ imho should be:

$\psi _{2}=Be^{ikx}\left({\begin{matrix}-1\\1\end{matrix}}\right),\quad k=V_{0}-E_{0}\,$

This leaves us with $A=0$, which is confusing, and $A'=B$, which is pair creation.

I believe real-valued wave functions are needed.

Please give your comment.

Aoosten (talk) 23:35, 15 December 2011 (UTC)

What is the nature of the propogating "positron" state?

It appears that, if I fire a wavepacket at this barrier, then the wavepacket will tunnel through just like the plane waves. On the far side of the barrier, its group velocity will be maintained. Also, due to charge conservation it will obviously have the same charge as the original wave (I have heard people say that Klein tunneling turns the electron into a positron (hole), but that seems suspicious). I suspect this has to do with some of the ambiguities about electron holes...

Does anyone know of some good research into this topic? --Nanite (talk) 13:42, 20 May 2013 (UTC)

Klein paradox for massive particles (suggestion for the article)

The classic treatment of the Klein paradox in Bjorken and Drell is wrong which resulted in a number of bogus papers and other textbooks that just copied the error. See for example this article by Wergeland and page 307 of this book by Thaller on the Dirac equation where it is stated that Klein himself made an important remark on the nature of the solution inside the step region which is ignored by Bjorken and Drell. The paper by Hansen that is cited in the section on the resolution on the massive case copies this error directly from Bjorken and Drell. They even acknowledge the error but refuse to make any changes. The second paper that is cited in this section contains the same spurious solution. The Klein paradox refers to finite transmission for potentials that exceed $E+mc^2$. In these papers it is claimed that no transmission will actually occur and that the "naive" calculation in the single-particle theory breaks unitarity which is apparently not a problem?! It is a problem and it is caused by matching to the wrong solution in the step region in the Klein paradox regime.

In the same section, it is claimed that the transmission reaches one in the massive case as the potential goes to infinity. This is not true. Instead for a step potential the correct calculation gives a maximum transmission of $2pc/(2pc+E)$ in the limit that the potential height goes to infinity. The paradox is completely solved by noting that previous negative energy states become positive states inside the step. Therefore there are positive energy propagating modes (these modes have to have the same energy as the incoming particle) inside the step to which an incident particle can propagate. The origin of the paradox is the unboundedness of the Dirac spectrum. You do not need quantum field theory at all to resolve the paradox.

The picture (Fig. 1) in the article is also wrong. The mode in the step region corresponding to the left leg of the cone has modes that propagate to the right since the slope is positive even though the momentum is negative (the phase velocity is negative but the group velocity is positive).

I suggest that this article is rewritten and only gives the calculation for the massive case for normal incidence at a step potential which was the original formulation by Klein, with emphasis that the treatment in a lot of textbooks (that just copied Bjorken and Drell) is wrong.

PraanWiki (talk) 15:25, 18 September 2015 (UTC)

You're right, the figures at first glance are in direct contradiction as one shows the k vector pointing right inside the step, whereas the other shows (correctly) that the k vector points to the left. I suppose they were trying to represent group velocity with the arrows, but it just adds confusion if a plane wave (not a wave group/packet) is shown.

You can (and should) be bold and update the article yourself! Since it's the Klein paradox, it certainly makes sense to discuss the original sense of this paradox in Klein's own terms. All of the future, overcomplicated/incorrect reinterpretations in terms of pair production can be relegated to a small section at the end. --Nanite (talk) 17:13, 18 September 2015 (UTC)