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Dirac_5 - kx i t E kx i L − − − − − Ψ = = = −...

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PHY4604 R. D. Field Department of Physics Dirac_5.doc University of Florida Negative Energy Solutions Klein-Gordon (Free Particle Solutions): The Klein-Gordon ( relativistic ) equation results in twice as many free particle solution than Schrodinger’s equation ( non-relativistic ). The two solutions with positive energy are fine ( right and left moving particles ). The negative energy solutions are a disaster! They lead to negative probability densities and since they are lower in energy particles could make a transition to the lower energy state. Solution Momentum Energy Interpretation ) / ( ) , ( h Et kx i R Ae t x + = Ψ k p x h = 2 2 2 ) ( ) ( mc cp E x + = Right moving particle ) / ( ) , ( h Et kx i L Ae t x + + = Ψ k p x h = 2 2 2 ) ( ) ( mc cp E x + = Left moving particle ) / ( ) , ( h Et kx i R Ae t x + = Ψ k p x h = 2 2 2 ) ( ) ( mc cp E x + = Unknown ) / ( ) , ( h Et kx i L Ae t x = Ψ k p x h = 2 2 2 ) ( ) ( mc cp E x + = Unknown Running Time Backwards: Notice that if we let t –t then ) , ( ) , ( ) / ( ) / ) ( ( t x Ae Ae t x R Et kx i t E kx i R + + Ψ = = = Ψ h h ) , ( ) , ( ) / ( ) / ) ( ( t x Ae Ae
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Unformatted text preview: kx i t E kx i L + + − − − − − Ψ = = = − Ψ h h If we run time backwards the negative energy solutions become positive energy solutions. Antiparticles: The prescription we use for handling the negative energy states was proposed by Stükelberg (1941) and Feynman (1948). The idea is that a negative energy solution describes a particle which propagates backward in time or, equivalently, a positive energy antiparticle propagating forward in time. Negative-energy particle solutions going backward in time describe positive-energy antiparticles going forward in time. Time e-photon Electron-Photon Scattering Space e-e-Time e-photon Electron-Positron Creation & Annihilation Space e + e-Time e-= e + E > 0 (-E) < 0 Feynman diagram!...
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