Dirac_9 - four differential equations which couple the four...

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PHY4604 R. D. Field Department of Physics Dirac_9.doc University of Florida The Dirac Equation (Covariant Form) Dirac Equation: We see that 2 mc p c H op β α + = r r , with t t r i t r H op Ψ = Ψ ) , ( ) , ( r h r , and hence ) , ( ) , ( ) , ( 2 t r mc t r c i t t r i r r r r h r h Ψ + Ψ = Ψ If we multiply by β we get ) , ( ) , ( ) , ( 2 t r mc t r c i t t r i r r r r h r h Ψ + Ψ = Ψ and hence 0 ) , ( 2 = Ψ + t r mc c i t i r r r h h . This can be written as 4-vector dot product as follows () ( ) 0 ) , ( ~ ~ ) , ( ~ ~ 2 2 = Ψ = Ψ t r mc i t r mc p r h r γ with z y x βα ~ and = = z y x t z y x c i c i c i i cp cp cp E p h h h h ~ . Dirac Gamma Matrices: The four 4 × 4 matrices γ 0 = β , γ 1 = βα x , γ 2 = βα y , γ 3 = βα z are referred to as the “Dirac Gamma Matrices” and the Dirac Equation is written as (p – mc 2 ) Ψ = 0 where we define A A ~ ~ for any 4-vector A ~ . The Dirac equation is really
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Unformatted text preview: four differential equations which couple the four components of a single column vector Ψ as follows: ) , ( ) , ( ) , ( ) , ( 1 1 1 1 3 2 1 2 = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ Ψ Ψ Ψ Ψ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⋅ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ t r t r t r t r mc c i c i c i i z y x t z y x r r r r h h h h Dirac Equation (covariant form)...
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