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Klein Paradox

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Klein Paradox

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    1. Klein Paradox

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

    3. An Overview

    4. 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) Same energy for same time dependenceSame energy for same time dependence

    5. Klein Paradox from KG eqn point of view In case of a potential, V E is kinetic energyE is kinetic energy

    6. Solutions in Region I:

    7. From KG eqn,

    9. To get the values of R,T apply continuity conditions of F and its derivative at z=0:

    10. Conservation of Charge Current Charge current is defined by:

    14. Dirac Equation and Klein Paradox For z<0 : Reflected and transmitted waves don necessarily have the same direction of spin. Spin is a form of energy and energy can be supplied or absorbed from the potential barrierReflected and transmitted waves don necessarily have the same direction of spin. Spin is a form of energy and energy can be supplied or absorbed from the potential barrier

    15. Reflected spinor:

    16. Consider the Case of a Strong Potential: By applying the continuity conditions of the spinors at the boundary:

    17. The Probability Currents:

    18. Where r =

    19. 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

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

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