I am reading this paper about the Klein paradox, i.e. transmission of relativistic particles incident on a potential step of height $V_0 > E + mc^2 > 2mc^2$ with $E$ the energy of the incident particle.

In most textbooks (e.g. Bjorken and Drell) you read that the paradox is resolved by considering the production of particle-antiparticle pairs at the potential step which can be naturally incorporated in quantum field theory.

For spinless bosons described by the Klein-Gordon equation, this picture seems satisfactory. An incident particle moving to the right towards the step annihilates with a left-moving antiparticle at the potential step and the partner particle is transmitted to the right.

However, for fermions, the paper seems to conclude that the incident particle is completely reflected and that no pair creation can occur because the reflected particle already occupies this mode. Also, naively, the single-particle calculation seems to violate unitarity.

This other paper seems to conclude the same, i.e. there is no transmission for fermions, just total reflection.


Is this the correct picture? Are the textbooks wrong on this one? Also, does pair creation at the potential violate energy conservation or is the energy supplied by the static potential?


I have found a very interesting discussion on page 307 of the book B. Thaller, The Dirac equation.

In 1928, O. Klein discovered oscillatory solutions inside a potential step where a nonrelativistic solution would decay exponentially. He determined the reflection and transmission coefficients for a rectangular step potential $V_0 \theta(x)$. Subsequently F. Sauter investigated Klein's paradox for a smooth potential, which gave the same qualitative result, but with a much smaller transmission coefficient. Klein's paradox is also described in the book of Björken and Drell, but in their "plane wave treatment" of the problem is a serious error, as was pointed out first by Dosch, Jensen and Muller. Björken and Drell considered a solution as "transmitted" which in fact corresponds to an incoming particle. Obviously, they neglected the fact that the velocity of the transmitted wave is opposite to its momentum (which is typical for negative energy solutions ... and was already noted by Klein). They concluded that more is reflected than comes in, which is incorrect and contradicts, e.g. the unitarity of time evolution. Unfortunately, the same error is contained also in many papers which treat that subject on a formal level.

Neither of the above papers are completely correct in my opinion. The conclusion is that one needs to take the correct solution with positive probability current in the Klein paradox regime. Note that there is still a Klein paradox which is just transmission through a huge potential and that unitarity is preserved.

  • $\begingroup$ recently i saw a work on graphene ;Nature Physics 2, 620 - 625 (2006) Published online: 20 August 2006 | doi:10.1038/nphys384 M. I. Katsnelson,K. S. Novoselov,A. K. Geim Chiral tunnelling and the Klein paradox in graphene does this help in clarifying the picture of "klein Paradox" M. I. Katsnelson1, K. S. Novoselov2 & A. K. Geim2 $\endgroup$ – drvrm Mar 2 '16 at 17:41

There are pro-forma classical solutions available mixing on one side of potential step particles with, on the other, antiparticles. Such situation implies the need for understanding of what is the physics problem one is solving. This situation was addressed and studied in depth in early-mid seventies and detailed studies were published as a part of a Physics Reports https://doi.org/10.1016/0370-1573(78)90116-3 and there is another presentation in this book https://www.amazon.com/Quantum-Electrodynamics-Strong-Fields-Introduction/dp/3642822746 . Alex Hansen and Finn Ravndal paper as I recall was an outsider view point. Another insight: Oscar Klein's "paradox" paper was sandwiched in-between the computation he was doing of the Klein-Nishina formula for Compton scattering BEFORE it was understood how to interpret E<-m solutions of the Dirac equation. Klein needed to clarify the role of these states as they were finding that all of the scattering was due to E<-m solutions. Thus before positrons were postulated, and before QED was formulated the formal solutions with coexistent outgoing (pair) waves were a (Klein's) paradox. I would say there is little paradoxical in this today. Giving paper clip answer how to deal with specific physics problems involving strong fields is not possible in these pages.

| cite | improve this answer | |

Really interesting Question. To me, this Seems like Quantum-mechanical-hydraulic jump. As it's mostly studied on fluid flows, it can't be very obvious why I parallel with this Klein-paradox; Hydraulic jump happens when Froude number is above one. (If $Fr=\sqrt3$ this jump is dissipative.)

When Froude Number is one. The inertia is in balance with external field when $Fr=1=\frac{v}{\sqrt{gl}}$,
which I prefere to write in from $1=\frac{v^2}{{gl}}$

If we change these values to Energy (or metric); we have another (metric)ratio; $1/2$
as Kinetic energy is $E_k=1/2mv^2$ and Potential energy is $E_p=mgl$ when $(l=h)$.

So what do we have in Klein paradox? $V_0$ is the total Potential step of height. $E$ is the kinetic energy of the particle and $mc^2$ is the mass energy of Particle, and as these have a rule
$V_0 > E +mc^2> 2 mc^2$, which is the paradox, as it's violating the conservation of the energy.

Let's take a closer look; as
$E +mc^2> 2 mc^2$ thus also
$E >mc^2$

So if the Froude Number would be one, we would have
$V_0 = E + 2 mc^2 = 3 mc^2$ thus we can conclude that
$V_0 > E +mc^2> 2 mc^2$ has obviously $Fr>1$,
more precisely it has $E /mc^2$ which means that the Metric ratio is $1/1$, which simply means $Fr=\sqrt2$


Is this the correct picture?

A: No.


Are the textbooks wrong on this one?

A: Yes.


Also, does pair creation at the potential violate energy conservation or is the energy supplied by the static potential?

A: Yes, it does violates the Energy conservation, or rather, as this can't be violated proves that the Current physical theories are wrong.

Here's a part of the page 120 of the book provided in Question;

Klein's Paradox

At the text there is word "kinetic Energy" and "External Field" underlined with red. This is done because I expect this answer to be deleted because many will think it "Doesn't answer the question". So please do give a minus vote, but don't delete simply because of the lack of understanding. -Thanks.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.