Klein paradox

Named after the Swedish physicist Oskar Klein, the Klein paradox is a property of relativistic quantum mechanics pertaining to the scattering of a wave function from a potential barrier. When the incoming energy of a particle is less than the height of the barrier, the particle should classically be reflected with 100% certainty. But the Klein-Gordon or Dirac equations have a classically spurious transmitted wave into the potential region, where the particle should classically not be able to go by energy conservation. In a quantum context, i.e., non-classically, the transmitted wave function solution physically describes propagation of an anti-particle of the originally incident particle. This physical interpretation agrees with experiment but precludes a single-particle interpretation of relativistic quantum mechanics. The resulting combination of quantum mechanics with special relativity without a single particle interpretation of a wave function at any given point leads to quantum field theory.

Although in a modern field theoretical interpretation of the Dirac equation the Klein paradox is automatically resolved, it continues to inspire publications today.

Using computer simulations they were able to show that the transmitted wave is the amount of suppression of pair creation at the barrier due to Pauli exclusion from the incoming electron. Except for a stream of positrons, there are no electrons under the barrier and the incoming electron is 100% reflected, although it gets entangled with the other electrons and positrons that are created at the barrier.

The Klein Paradox in 1D

Below is an analytical representation of the Klein Paradox. It is based on the probability current definition in Quantum Mechanics, and results in a nonzero transmission coefficient, even at large potential barriers.

Assume a step potential barrier as illustrated:

We assume no incoming waves from the right side.

The Dirac Equation is:

$E\psi =H\psi \,$

Explicitly:

$\left(\alpha p+\beta m+V\right)=E\psi$

Where:

$V={\begin{cases}0&x<0\\V_{0}&x>0\end{cases}}\,$

We choose the matrices to be

$\alpha =\sigma _{x},\ \beta =\sigma _{z}\,$

The eigenvectors obtained are:

$\psi _{1}=Ae^{ikx}\left({\begin{matrix}k\\E-m\end{matrix}}\right)+A'e^{-ikx}\left({\begin{matrix}-k\\E-m\end{matrix}}\right)\,$

$\psi _{2}=Be^{ipx}\left({\begin{matrix}p\\V_{0}-E+m\end{matrix}}\right)\,$

The definition of probability current is:

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

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

In this case:

$J_{1}=2k\left(E-m\right)\left[\left|A\right|^{2}-\left|A'\right|^{2}\right]\,$

$J_{2}=2\left|B\right|^{2}\left(V_{0}-E+m\right)p\,$

Now define:

$J_{1}=j_{i}-j_{r},J_{2}=j_{t}\,$

and

$R={\frac {j_{r}}{j_{i}}}\,$

$T={\frac {j_{t}}{j_{i}}}\,$

This yields:

$R={\frac {\left|A'\right|^{2}}{\left|A\right|^{2}}}$

$T={\frac {p}{k}}\cdot {\frac {V_{0}-E+m}{E-m}}\cdot {\frac {\left|B\right|^{2}}{\left|A\right|^{2}}}$

We define

$\kappa ={\frac {p}{k}}\cdot {\frac {V_{0}-E+m}{E-m}}$

And, using continuity at $x=0$ we get:

$R=\left|{\frac {\kappa -1}{\kappa +1}}\right|^{2}$

$T={\frac {4\kappa }{\left|\kappa +1\right|^{2}}}$

Now, we use the known relations:

$k={\sqrt {E^{2}-m^{2}}}\,$

$p={\sqrt {\left(V_{0}-E\right)^{2}-m^{2}}}\,$

Plugging this in, we can see that for massless particles, $\kappa =1$, and so $R=0,T=1$ for every potential. Also, for massive particles, as the potential gets bigger, we get:

$V_{0}\rightarrow \infty ,\kappa \not \rightarrow 0$

And so even as the potential step rises to infinity, there always exists a nonzero transmission.

Links

• Klein Paradox for Spin-0 Particles
• Klein Paradox for Spin-1/2 Particles
• Klein Paradox