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Quiz1 - df/dx in the form df dx = A B)exp xA)exp xB[exp...

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PHY4221 Quantum Mechanics I (Fall 2004) Quiz 1 Provided that two operators A and B which do not commute with each other but commute with their commutator [ A, B ], i.e. C [ A, B ] 6 = 0 and [ A, C ] = [ B, C ] = 0 , verify the Baker-Campbell-Hausdorff relation: exp ( A ) exp ( B ) = exp A + B + 1 2 [ A, B ] . Hint : Start with the function f ( x ) = exp( xA ) exp( xB ) and determine the first derivative
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Unformatted text preview: df/dx in the form df dx = ( A + B )exp( xA )exp( xB ) + [exp( xA ) ,B ]exp( xB ) and show that [exp( xA ) ,B ] = x [ A,B ]exp( xA ) leads to the first-order differential equation df dx = ( A + B + x [ A,B ]) f ( x ) giving after integration the Baker-Campbell-Hausdorff relation....
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