KleinParadox

The Klein paradox is a counterintuitive phenomenon in quantum mechanics which shows that relativistic quantum particles can penetrate potential barriers, even those of arbitrary height and thickness. Named after Swedish physicist Oskar Klein, the phenomenon was first encountered in the context of quantum electrodynamics (QED) but also manifests in quantum field theory.

Let's discuss this paradox in the context of the Dirac equation, which describes relativistic particles like electrons.

The one-dimensional Dirac equation in an electrostatic potential can be written as:

$$ i\hbar \frac{\partial \psi}{\partial t} = c\sigma_x p + m_0c^2\sigma_z + e\phi(x)\psi $$

where:

• $\hbar$ is the reduced Planck's constant
• $\psi$ is the Dirac spinor
• $c$ is the speed of light
• $\sigma_x$ and $\sigma_z$ are the Pauli matrices
• $p$ is the momentum operator
• $m_0$ is the rest mass of the particle
• $e$ is the elementary charge
• $\phi(x)$ is the electrostatic potential

Let's say we have a step potential, which is given by:

$$ \phi(x) = \begin{cases} V_0 & \text{if } x > 0 \\ 0 & \text{if } x < 0 \end{cases} $$

The Dirac equation for the two regions (x < 0 and x > 0) then separates into two equations. Solving these two equations provides the solutions for the incident, reflected, and transmitted waves.

The transmission coefficient $T$, which describes the probability of the particle tunneling through the potential barrier, becomes:

$$ T = \frac{1}{1 + Z^2} $$

Where:

$$ Z = \frac{(E - V_0)^2 - m^2c^4}{2mV_0(E + mc^2)} \quad \text{(for } V_0 > E\text{)} $$

Here $E$ represents the total energy of the particle, $m$ represents the relativistic mass and $V_0$ is the potential step height. According to the Klein paradox, for a potential step higher than twice the rest mass energy of the particle (i.e., $V_0 &gt; 2m_0c^2$), the transmission probability $T$ increases, even becoming 1 for an infinitely high potential step. This is completely counterintuitive from the point of view of non-relativistic quantum mechanics where, for a potential higher than the energy of the particle, there is no transmission.

This phenomenon, while paradoxical, does not contradict relativity or quantum mechanics and is a consequence of the coupling of positive and negative energy states in the Dirac equation, a necessary feature for the description of particle-antiparticle creation and annihilation processes. In more tangible terms, this reflects that when a particle meets an energy barrier of sufficient height, it can create a particle-antiparticle pair, with the antiparticle moving backwards and the new particle moving forwards, thus seeming to move through the barrier.