ScatteringOfLight

ScatteringOfLight - Notes on atom-photon scattering: The...

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Unformatted text preview: Notes on atom-photon 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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ScatteringOfLight - Notes on atom-photon scattering: The...

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