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Dirac_10 - 2 and eigenvectors ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞...

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PHY4604 R. D. Field Department of Physics Dirac_10.doc University of Florida Free-Particle Spinors Free-Particle Solutions: Look for solutions of the Dirac equation (p – mc 2 ) Ψ( x,t ) = 0 of the form c x p i e p u t r h r r / ~ ~ ) ( ) , ( = Ψ . Since we are seeking the energy eigenvalues it is easier to use the original form of the Dirac equation as follows: t t r i t r H op Ψ = Ψ ) , ( ) , ( r h r with 2 mc p c H op β α + = r r . Substituting in Ψ from above yields ( ) ) ( ) ( ) ( 2 p Eu p u mc p c p u H op r r r r r = + = β α . There are four independent solutions of this equation, two with E > 0 and two with E < 0. Free-Particle at Rest: If we take p = 0 and use the Dirac-Pauli representation then we see that ) 0 ( ) 0 ( ) 0 ( 2 Eu u mc u H op = = β . In this case, ) 0 ( ) 0 ( 0 0 0 0 0 0 0 0 0 0 0 0 ) 0 ( 2 2 2 2 2 Eu u mc mc mc mc mc u H op = = with eigenvalues E = mc 2 , mc 2 , -mc
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Unformatted text preview: 2 and eigenvectors ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 1 ) ( 1 u , ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 1 ) ( 2 u , ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 1 ) ( 3 u , ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 1 ) ( 4 u . The electron charge –e is regarded as the “particle”. Therefore the first two solutions (with E > 0) describe the two spin states of the electron and the last two solutions (with E < 0) describe the two spin states of the positron ( i.e. antiparticle with charge +e)....
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