A positive-energy Dirac packet crosses a scalar barrier higher than its incident energy; normal incidence can remain strikingly transparent in the Klein zone.
This page describes the phenomenon using the current QPE preset and numerical modules as its primary source.
The Klein branch is deliberately formulated as a two-component effective Dirac model. The QPE barrier does not create particle pairs: when V₀ crosses the gap thresholds, the barrier interior supports propagating states on the other branch and transmission can re-emerge instead of monotonically decreasing.
The paradox. QPE prepares a positive-energy Dirac packet and sends it toward a scalar barrier V0. With the defaults E is about 1.50 while V0=3.2, far above the incident energy.
Press Play. Analysis -> Klein Tunneling tracks probability left of, inside, and right of the barrier. At normal incidence a large fraction can cross the supercritical region.
Why. When V0 > E + mc^2, the relative energy E-V0 lies in the negative-energy continuum. The barrier is no longer merely an evanescent forbidden region: the opposite Dirac branch can propagate inside it and connect to a transmitted state.
Try the angle. Increase py and press Apply / Reset. Oblique incidence suppresses transmission; near-normal incidence gives the most striking result.
History. Oskar Klein discussed the apparently paradoxical high-potential behaviour of the Dirac equation in 1929. Klein tunneling later became especially important in effective Dirac systems such as graphene and quantum simulators.
Model boundary. This step evolves a two-component 2D Dirac equation in an external scalar potential. It is not QED: no real pair creation or dynamical electromagnetic field is included. Positive/negative branches refer to the single-particle/effective-pseudospin Dirac model used by the Explorer.
Oskar Klein analysed relativistic reflection from a potential step in 1929. The paradox became experimentally accessible in effective systems such as graphene, where chiral quasiparticles mimic Dirac fermions with velocity far below c.
| Representative platform | gapped effective Dirac quasiparticle / graphene-like simulator |
|---|---|
| Mapping basis | effective-Dirac velocity mapping (v_D~10^6 m/s) |
| 1 QPE length unit | 10 nm |
| 1 QPE time unit | 10 fs |
| Energy scale E0=ħ/T0 | 65.8 meV |
| Equivalent frequency E0/h | 15.9 THz |
| Note | Uses an effective Dirac velocity near 10^6 m/s, so one 10 nm length unit maps to about 10 fs. The nonzero QPE mass is an effective gap parameter, not the vacuum electron rest mass; this is not a material-specific graphene simulation. |
Open Model & Constraints to see exactly what QPE computes and what is left outside the model. For current live values use Help → Current Experiment → Current Parameters.