Dirac_6 - = 1 1 x (Pauli matrix) and = 1 1 I (identity...

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PHY4604 R. D. Field Department of Physics Dirac_6.doc University of Florida The Dirac Equation (One Dimension) For some time it was thought that the Klein-Gordon equation was the only relativistic generalization of Schrödinger’s equation until Dirac discovered an alternative one. His goal was to write an equation, unlike the Klein- Gordon equation, that is linear in t / . One-Dimensional Dirac Equation: We look for a Hamiltonian of the form 2 mc cp H x x op β α + = , with t t x i t x H op Ψ = Ψ ) , ( ) , ( h . The constants α x and β are determined by requiring 2 2 2 2 ) ( ) ( mc cp H x op + = . We see that () () ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 ) ( ) ( ) ( ) ( ) ( mc cp mc cp mc cp mc cp mc cp mc cp x x x x x x x x x x x x + = + + + = + + = + βα Hence, 1 2 2 = = x 0 } , { = + = x x x Clearly α x 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 matrices α x and β are not unique. One choice is = 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 x x x σ and = I I 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 where
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Unformatted text preview: = 1 1 x (Pauli matrix) and = 1 1 I (identity matrix). We see that = = = I I x x x x x x x 2 2 2 = = I I I I I I 2 = + = + = + x x x x x x x x x x I I I I...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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