Klein Paradox

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

This Demonstration shows the reflection and transmission coefficients for a Dirac particle with spin-up incident on a square barrier of variable height The energy of the particle is fixed at 1 unit ), but its mass is allowed to vary from 0 to . What we see is that the negative energy solutions of the Dirac equation become accessible on the barrier side, thus allowing both reflection and transmission coefficients to increase beyond 1: Klein's paradox! The phenomenon can be explained by particle-antiparticle pair creation at the potential barrier.

Contributed by: Niels Walet (February 2010)
Open content licensed under CC BY-NC-SA



[1] O. Klein, "Die Reflexion von Elektronen an Einem Potentialsprung Nach der Relativistischen Dynamik von Dirac," Z. Phys., 53(3-4), 1929 pp. 157–165.

[2] See any textbook on relativistic quantum mechanics.

[3] Wikipedia entry for Klein paradox.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.