Klein Paradox - Massless Particles

Massless Particles

Consider a massless relativistic particle approaching a potential step of height with energy and momentum .

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

And is the Pauli matrix:

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):

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, They are derived from the probability amplitude currents.

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

In this case:

The transmission and reflection coefficients are:

Continuity of the wave function at, yields:

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.