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# Dirac_6 - PHY4604 R D Field The Dirac Equation(One...

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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 β α β α β α β α β α Hence, 1 2 2 = = β α x 0 } , { = + = x x x βα β α β α 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.
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