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# Dirac_11 - PHY4604 R D Field Free-Particle Spinors(p 0...

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PHY4604 R. D. Field Department of Physics Dirac_11.doc University of Florida Free-Particle Spinors (p 0) Free-Particle Solutions: In the Dirac-Pauli representation and p 0 the free particle equation ( ) ) ( ) ( ) ( 2 p Eu p u mc p c p u H op r r r r r = + = β α . becomes = B A B A u u E u u mc p c p c mc 2 2 r r r r σ σ , where the 4-component spinor ) ( p u r has been divided into two 2-component spinors u A and u B as follows: = B A u u p u ) ( r . We see that (1) A B u mc E u p c ) ( 2 = r r σ and (2) B A u mc E u p c ) ( 2 + = r r σ . For the two E > 0 solutions we can take = = + 0 1 ) 1 ( χ A u and = = + 1 0 ) 2 ( χ A u . The “lower components” of u are determined from (2) above as follows: + + = χ σ 2 ) 1 ( mc E p c u B r r and + = χ σ 2 ) 2 ( mc E p c u B r r . Hence, the two positive energy (E > 0) 4-spinor solutions are + = + + > χ σ χ 2 ) 1 ( 0 ) ( mc E p c N p u E r r r and + = > χ σ χ 2 ) 2 ( 0 ) ( mc E p c N p u E r r r , where N is the normalization constant.
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