# Klein Paradox - Dirac equation with step potentional

• Drew Carey
In summary, the conversation discusses the solution to the Klein paradox in Itzykson & Zuber's Quantum Field Theory, specifically the solution to the Dirac equation with a step potential. The solution is based on solving for two regions and requiring continuity at the origin, but does not enforce continuity of the first derivative. This is different from the approach typically used in solving Schrodinger's equation with a step potential. The question arises as to why the continuity of the first derivative is not necessary in the Dirac equation, given that it is supposed to be equal to Schrodinger's equation in the non-relativistic limit. The response is that Dirac's equation provides extra information and the non-relativistic limit simplifies the spinors
Drew Carey
Hi all!

I was reading up on the Klein paradox in Itzykson & Zuber's Quantum Field Theory (but I think this is a pretty standard part that's probably present in most QFT textbooks) and on page 62 they have a pretty straight forward solution to the Dirac equation with a step potential.

I've noticed something odd though - they're solution is based upon solving for the two regions (V=0 and V>0) and requiring continuity at the origin. They do not however require continuity of the first derivative, and in fact their solution's first derivative isn't continuous.

When solving Schrodinger's equation with a step potential it is always customary to enforce continuity of the first derivative as the equation contains a second space derivative.

I've tried to solve the Dirac problem myself with this added constraint, but is seems unsolvable (over constrained. Though I may have some mistake in the algebra). I've searched online for other solutions to this problem, and they all seem to ignore continuity of the first derivative, without even mentioning it.

What am I missing? How can we give up on continuity of the first derivative? Especially in light of the fact that the Dirac equation is supposed to be equal to Schrodinger's equation in a non-relativistic limit these two constraints need to be satisfied by the Dirac equation solution as well, no?

Last edited:
Why do you take the continuity of the first spatial derivative for the Shrodinger's equation?
Because it's a 2nd order differential equation (wrt spatial components).
This is of course not the case for Dirac's equation, where you only have one spatial derivative.
$(\slash{p} \pm m) \psi=0$

The fact that Dirac's equation in the NRL yields the classical results, should not be confused of that fact that it gives even more extra information over physics. For example the non relativistic limit will drop your spinors from 4 to 2 component, while in fact they exist in one single 4component spinor.

## 1. What is the Klein Paradox?

The Klein Paradox is a phenomenon in quantum mechanics where a particle with high enough energy can penetrate a potential barrier that would normally be impenetrable. It was first described by physicist Oskar Klein in 1929.

## 2. What is the Dirac equation?

The Dirac equation is a relativistic wave equation that describes how particles with spin, such as electrons, behave in the presence of an electromagnetic field. It was developed by physicist Paul Dirac in 1928 and is a fundamental equation in quantum mechanics.

## 3. What is a step potential?

A step potential is a sudden change in the potential energy of a particle. In the context of the Klein Paradox, it refers to a potential barrier that is infinitely high on one side and zero on the other side. This creates a sudden jump in potential energy for a particle moving through it.

## 4. How does the Klein Paradox relate to the Dirac equation?

The Klein Paradox is a consequence of the Dirac equation. When a particle with high enough energy encounters a step potential, the Dirac equation predicts that it will have a non-zero probability of being reflected back instead of being completely blocked by the barrier. This is due to the particle's high energy allowing it to behave like an anti-particle, which can move through the barrier with ease.

## 5. Can the Klein Paradox be observed in real-life experiments?

Yes, the Klein Paradox has been observed in experiments with high-energy particles, such as electrons and positrons, moving through a potential barrier. This phenomenon has also been observed in other fields of physics, such as in optics with high-energy photons. However, it is still a rare occurrence and requires specific conditions to be met in order to be observed.

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