Getting Started with Klein's Paradox: A Forum Q&A

[plxmny]

I am interested in learning more about Klein's paradox. My QM book said you need QFT to treat it properly. I did a quick internet search but found no expository materials. Even Wikipedia had no mention. This forum had very little of substance that I could find on the subject.

Maybe someone can get me on the right track?

 

[meopemuk]

I am interested in learning more about Klein's paradox. My QM book said you need QFT to treat it properly. I did a quick internet search but found no expository materials. Even Wikipedia had no mention. This forum had very little of substance that I could find on the subject.

Maybe someone can get me on the right track?

The Klein's paradox is related to non-conservation of probabilities (interpreted as "probability currents") when Dirac's equation is used for calculating the electron scattering on potentials. You can find the description of this paradox in J. D. Bjorken and S. D. Drell "Relativistic quantum mechanics" (1964), p. 40-42.

In my opinion, this "paradox" is simply an indication that Dirac's equation is not a valid relativistic analog of the Schroedinger equation for electrons, and that Dirac's "wavefunction" does not have a probabilistic interpretation. It is more consistent to describe relativistic quantum problems in terms of the Schroedinger equation

$$i \hbar \frac{\partial \psi(\mathbf{r}, t) }{\partial t }= (\sqrt{- \hbar^2 c^2 \nabla^2 + m^2c^2} + V(\mathbf{r}) )\psi(\mathbf{r},t)$$

with $|\psi(\mathbf{r},t)|^2$ interpreted as the probability density. You can find additional discussions of these points among recent posts in the thread https://www.physicsforums.com/showthread.php?t=175155

You will not find discussions of the Klein's paradox in QFT, because this theory does not describe scattering as a time-dependent process controlled by the Schroedinger equation. Instead, QFT calculates directly the S-matrix (the mapping of asymptotic states in the remote past to asymptotic states in the remote future). If everything is done correctly (renormalization, etc.), then the S-matrix is unitary, which means that squares of absolute values of the S-matrix elements can be interpreted as probabilities (scattering cross-sections) and the sum of all probabilities is equal to 1, as it should be.

Eugene.

 

[olgranpappy]

The "Klein Paradox" and other weird effects such as "Zitterbewegung" are kind of interesting, but they are probably not real effects. As Eugene said, they indicate a failure of the single-particle theory.

There are similar effects, though, that can be observed in graphene (single layer of graphite) because in graphene the electron's energy momentum relation turns out to look just like that of a massless particle. And in graphene there really is a working single particle theory and there really is a "filled fermi sea." So you can observe a real "Klein effect" that is somewhat analogous to the Klein paradox... but not really.

Anyways, there actually has been some recent work regarding whether or not there really is a Klein paradox-type effect which concludes (roughly) that there is not. The reference is:

Phys. Rev. Lett., 92, 040406

 

[meopemuk]

The "Klein Paradox" and other weird effects such as "Zitterbewegung" are kind of interesting, but they are probably not real effects. As Eugene said, they indicate a failure of the single-particle theory.

To be more precise, in my opinion, these "effects" indicate a failure of the single particle theory based on the Dirac equation. The single-particle theory based on the relativistic Schroedinger equation (that I wrote) is just fine, as long as energies are not too high for the particle creation processes to take place.

Eugene.

 

[olgranpappy]

okay... but, the point of the Klein paradox is that the equation you wrote is *not* fine when V > 2mc^2.

 

[meopemuk]

okay... but, the point of the Klein paradox is that the equation you wrote is *not* fine when V > 2mc^2.

Are you sure we are talking about the same "Klein paradox"? A described in Bjorken and Drell, Klein paradox refers to the scattering of an electron on a potential. It appears that the probabilities of passed and reflected waves do not add up to 1. This is a serious violation of probability laws.

I suspect that you are talking about a different effect, which is characteristic to the (stationary) Schroedinger equation

$$(\sqrt{- \hbar^2 c^2 \nabla^2 + m^2c^2} + V(\mathbf{r}) ) \psi(\mathbf{r})= E \psi(\mathbf{r})$$

Suppose that $V(\mathbf{r})$ is a strong attractive potential (for example, a potential from a nucleus with a high charge Z). If the energy of the lowest bound state for an electron in this potential becomes lower than $-mc^2$ (which is $2mc^2$ lower than the energy $mc^2$ of an electron at infinity), then it becomes energetically possible to create and electron-positron pair out of vacuum, so that the electron gets attached to the potential and the free positron moves away. As far as I know, this effect has not been observed experimentally, but I don't think there is anything paradoxical in this situation. I believe that such processes of spontaneous electron-positron pair creation in strong fields are quite possible.

Eugene.

 

[olgranpappy]

here's a cute little link:

http://www.phys.uAlberta.ca/~gingrich/phys512/latex2html/node35.html

he also discusses the klein paradox for spin 1/2

 

[meopemuk]

here's a cute little link:

http://www.phys.uAlberta.ca/~gingrich/phys512/latex2html/node35.html

he also discusses the klein paradox for spin 1/2

Thank you for the link. It pretty much repeats what is written in Bjorken and Drell, although it uses the KG equation instead of the Dirac one. When I wrote the last post I didn't have B&D book in hand. Now I have it. You are right that Klein paradox appears in situations when the potential is strong. Then, as I wrote in the previous post, the single-particle picture is not adequate anymore. There is a possibility of spontaneous creation of electron-positron pairs, which should be taken into account. In such situations, no one-particle theory would work: neither KG, nor Dirac, nor Schroedinger equation. One needs to use the QFT formalism which explicitly allows for creation and destruction of particles.

Eugene.