W02L05_Dirac Algibra

W02L05_Dirac Algibra - Physics 637 2013F Name Pauli and...

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Physics’637 ’ ’ 2013F’ ’ Name___________________________’ Pauli’and’Dirac’Alge bra’ 1 1. Pauli Trace Algebra: ’ Here’I’will’use’the’notation’that’for’a’3 -vector’ v ’ ,’ The’bold’ face’indicates’the’combination’with’sigma’matrices: v = v σ ’ and’ 1 2 ’ is’the’identity’ 2 × 2 ’ matrix.’ a)’ Evaluate’ Tr ( a ) ,’ Tr ( ab ) ’ and’ Tr ( abc ) ’ in’terms’of’dot’products,’cross’products’ and/or’epsilon’tensors. ’ Tr ( a ) = 0 because’the’pauli’matrices’are’traceless. Tr ( ab ) = Tr ( a b 1 2 + i ( a × b ) ) = 2 a b Tr ( abc ) = Tr ( a b c + i ( a × b ) c ) = Tr ( i ( a × b ) c ) = 2 i ( a × b ) c ( ) [using previous result] b)’Show’that’ ab + ba = 2 a b 1 2 ’ ’ ab + ba = a b 1 2 + i ( a × b ) ( ) + a b 1 2 + i ( b × a ) ( ) = 2 a b 1 2
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W02L05_Dirac Algibra - Physics 637 2013F Name Pauli and...

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