In 1929, physicist Oskar Klein[1] obtained a surprising result by applying the Dirac equation to the familiar problem of electron scattering from a potential barrier. In nonrelativistic quantum mechanics, electron tunneling into a barrier is observed, with exponential damping. However, Klein’s result showed that if the potential is on the order of the electron mass, $V\sim mc^2$, the barrier is nearly transparent. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted.

## Massless particles

Consider a massless relativistic particle approaching a potential step of height $V_0$ with energy $E_0 and momentum $p$.

The particle's wave function, $\psi$, follows the time-independent Dirac equation:

$\left( \sigma_x p + V \right) \psi = E_0\psi,\quad V=\begin{cases} 0, & x<0 \\ V_0, & x>0 \end{cases}$

And $\sigma_x$ is the Pauli matrix:

$\sigma_x = \left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right)$
Fig. 1 A depiction of the dispersion relation, the x-axis represents momentum while the y-axis represents energy.

Assuming the particle is propagating from the left, we obtain two solutions — one before the step, in region (1) and one under the potential, in region (2):

$\psi_1=Ae^{ipx}\left( \begin{matrix} 1 \\ 1\end{matrix} \right)+A'e^{-ipx}\left( \begin{matrix} -1 \\ 1\end{matrix} \right) ,\quad p=E_0 \,$
$\psi_2=Be^{ikx}\left( \begin{matrix} 1 \\ 1\end{matrix} \right) ,\quad \left|k\right|=V_0-E_0 \,$

Where the coefficients A, A′ and B are complex numbers. Both the incoming and transmitted wave functions are associated with positive group velocity (Blue lines in Fig.1), whereas the reflected wave function is associated with negative group velocity. (Green lines in Fig.1)

We now want to calculate the transmission and reflection coefficients, $T, R.$ They are derived from the probability amplitude currents.

The definition of the probability current associated with the Dirac equation is:

$J_i=\psi_i^\dagger \sigma_x \psi_i,\ i=1,2 \,$

In this case:

$J_1=2\left[\left| A \right|^2-\left| A' \right|^2\right], \quad J_2=2\left| B \right|^2 \,$

The transmission and reflection coefficients are:

$R=\frac {\left|A'\right|^2} {\left|A\right|^2}, \quad T=\frac {\left|B\right|^2} {\left|A\right|^2} \,$

Continuity of the wave function at $x=0$, yields:

$\left|A\right|^2=\left|B\right|^2 \,$
$\left|A'\right|^2=0 \,$

And so the transmission coefficient is 1 and there is no reflection.

One interpretation of the paradox is that a potential step cannot reverse the direction of the group velocity of a massless relativistic particle. This explanation best suits the single particle solution cited above. Other, more complex interpretations are suggested in literature, in the context of quantum field theory where the unrestrained tunnelling is shown to occur due to the existence of particle–antiparticle pairs at the potential.

## Other cases

For the massive case, the calculations are similar to the above. The results are as surprising as in the massless case. The transmission coefficient is always larger than zero, and approaches 1 as the potential step goes to infinity.

These results were expanded to higher dimensions, and to other types of potentials, such as a linear step, a square barrier, etc. Many experiments in electron transport in graphene rely on the Klein paradox for massless particles.[2][3]