Abstract
We figure out the famous Klein’s paradox arising from the reflection problem when a Dirac particle encounters a step potential with infinite width. The key is to piecewise solve Dirac equation in such a way that in the region where the particle’s energy E is greater (less) than the potential V, the solution of the positive (negative) energy branch is adopted. In the case of Klein–Gordon equation with a piecewise constant potential, the equation is decoupled to positive and negative energy equations, and reflection problem is solved in the same way. Both infinitely and finitely wide potentials are considered. The reflection coefficient never exceeds 1. The results are applied to discuss the transmissions of particles with no mass or with very small mass.
Highlights
After Dirac equation was proposed, Klein came up with his famous paradox [1]
In the case of Klein–Gordon equation with a this work must maintain piecewise constant potential, the equation is decoupled to positive and negative energy equations, and attribution to the author(s) and the title of reflection problem is solved in the same way
The main content of this paradox was that when a relativistic particle encountered an infinitely wide potential barrier with height exceeding a certain value, the reflection coefficient would be larger than 1 [2]
Summary
After Dirac equation was proposed, Klein came up with his famous paradox [1]. Let us see the case when potential is absent It is solved from equation (1.1) that a free particle is of energy. Equation (1.7b) is applicable in regions where E < V, and it is called decoupled Klein–Gordon equation of negative energy branch. The clarification of the relationship between a particle’s energy and potential it is in helps one to correctly use the equation and corresponding wave function for both Dirac equation and decoupled Klein–Gordon equations. Both Dirac equation and Klein–Gordon equation are considered. The present work shows the necessity to use the solutions belonging to the negative energy branch in regions where a particle’s energy E is less than potential V.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call