Klein Paradox

Contents An Overview Klein Paradox and Klein Gordon equation Klein Paradox and Dirac equation Further investigation

An Overview

Assumptions We deal with a plane wave solution for KG, Dirac equation of a single particle The particles has an energy E (kinetic +rest energy) before and after the barrier A step-function potential Probability Current or charge current conserved No particle flux in Region II may come from the positive direction (causality requirement)

Klein Paradox from KG eqn point of view In case of a potential, V KG equation

Solutions in Region I: In region II:

From KG eqn, Weak potential P’ is real, Only positive value allowed Intermediate potential P’ is imaginary [Evanescent wave] Strong potential P’ is real A non-classical Behavior !!

What about charge current conservation?

To get the values of R,T apply continuity conditions of Φ and its derivative at z=0: Solving for R,T :

Conservation of Charge Current Charge current is defined by: This is interpreted as charge current not probability current since we have no positive definite conserved probability in KG equation

Transmission coefficient (T)= [ Logical result ] Reflection coefficient ( R )= In all cases T+R=1

For weak potential R=, T=, T+R=1 For an intermediate potential R=1, T=0, T+R=1 For a strong potential R=, T=, T+R=1 R>1, T is negative This is Klein paradox

 Reflected current is bigger than incident current !!!!!!  Transmitted current is opposite in charge to incident current !!!!!  Extra particles supplied by the potential ?  Another type of particle of opposite charge supplied by the potential ? Particle anti-particle pair production

Dirac Equation and Klein Paradox For z<0 : Incident wave: ( having spin up) For z>0 :

Reflected spinor: Transmitted spinor:

Consider the Case of a Strong Potential: By applying the continuity conditions of the spinors at the boundary: No spin flip

The Probability Currents:

Where r = = R= T= >1 Again reflected current is greater than incident current.

Hole Theory Explanation The potential energy raised a negative energy electron to a positive energy state creating a positive hole (positron) behind it. The hole is attracted towards the potential while the electron is repelled far from it !! This process is stimulated by the incoming electron

Question about this interpretation: How can energy conservation be guaranteed? Any experimental evidence !!!??? Should we reinterpret the probability current as charge current?

Further investigation To construct a wave packet of several momentum components and study its transmission and reflection behavior from a potential step. To use second quantization representation of dirac spinors

The End