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Unformatted text preview: Notes on atomphoton scattering: The Schrodinger equation for an electron in a potential V ( r ) interacting with quantized EM radiation is: i h t  i = H i (1) where H = 1 2 m p e c A ( r ) 2 + V ( r ) + X k , h k a k a k + 1 2 (2) In the transverse gauge, the gauge field operator operator A ( r ) = 1 V X k , s 2 h k c a k , e i k r + a k , * e i k r where [ a k , a k ] = kk ; [ a k , a k ] = 0 Now, let H = H + H (3) where H = H ( at ) + H ( rad ) = p 2 2 m + V ( r ) ! + X k , h k a k a k + 1 2 (4) and H = e mc A ( r ) p + e 2 2 mc 2 A ( r ) A ( r ) (5) In the interaction representation, we let  ( t ) i = e i h H t  ( t ) i . Note that in the absence of the perturbation H , the wavefunction  ( t ) i would be time independent. Subsequently i h t  i = H I ( t )  i = e i h H t H e i h H t  i = e i h H ( at ) t e i h H ( rad ) t H e i h H ( rad ) t e i h H ( at ) t  i Explicitly, we have H I ( t ) = e i h H ( at ) t e i h H ( rad ) t H e i h H ( rad ) t e i h H ( at ) t (6) = e i h H ( at ) t  e mc A ( r ,t ) p + e 2 2 mc 2 A ( r ,t ) A ( r ,t ) ! e i h H ( at ) t (7) where the time dependent gauge field operator is A ( r ,t ) = 1 V X k , s 2 h k c a k , e i k r i k t + a k , * e i k r + i k t Now, to 2 nd order in time dependent perturbation theory  ( t ) i  i + 1 i h Z t dt H I ( t )  i + 1 ( i h ) 2 Z t dt Z t dt 00 H I ( t ) H I ( t 00 )  i where we assumed that the perturbation is slowly switched on at t = ....
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 Spring '08
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