Dirac_7 - ⎞ ⎜ ⎜ ⎝ ⎛ = y y y ⎟ ⎟ ⎠ ⎞ ⎜...

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PHY4604 R. D. Field Department of Physics Dirac_7.doc University of Florida The Dirac Equation (Three Dimensions) Three-Dimensional Dirac Equation: We look for a Hamiltonian of the form 2 mc p c H op β α + = r r , with t t r i t r H op Ψ = Ψ ) , ( ) , ( r h r . where z z y y x x p p p p α α α α + + = r r . The four constants α x , α y , α z and β are determined by requiring 2 2 2 2 2 2 2 2 2 ) ( ) ( ) ( ) ( ) ( ) ( mc cp cp cp mc cp H z y x op + + + = + = . We require that ( ) ( )( ) 2 2 2 3 2 2 2 2 2 2 2 2 ) ( ) ( ) ( ) ( ) ( mc cp mc p p mc p c mc p c mc p c mc p c + = + + + = + + = + r r r r r r r r r r r r α β β α β α β α β α β α which implies 1 2 2 2 2 = = = = β α α α z y x 0 } , { = β α x , 0 } , { = β α y , 0 } , { = β α z 0 } , { = y x α α , 0 } , { = z x α α , 0 } , { = z y α α Clearly α x , α y , α z and β cannot be numbers. They must be matrices. The lowest dimensionality that meets the requirements are 4×4 matrices. The Dirac-Pauli Representation: The five (4×4) matrices α r and β are not unique. One choice is = 0 0 σ σ α r r r and = I I 0 0 β (Dirac-Pauli Representation)
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Unformatted text preview: ⎞ ⎜ ⎜ ⎝ ⎛ = y y y , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = z z z and where ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 1 x , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = i i y , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = 1 1 z are the Pauli matrices and ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 1 I is the identity matrix. The Weyl Representation: Another choice is ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− = r r r and ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = I I (Weyl Representation) ....
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