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md2011-04-note_Part_4

# md2011-04-note_Part_4 - As a model lets consider two...

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I As a model, let’s consider two oscillators, which are separated by distance R . I Each oscillator has charges of ± e with separations x 1 and x 2 . The particles oscillate along the x axis. + - + - x 2 x 1 R I Let p 1 and p 2 denote the momenta. I The force constant is C . I The Hamiltonian of the unperturbated system is H 0 = 1 2 m p 2 1 + 1 2 Cx 2 1 + 1 2 m p 2 2 + 1 2 Cx 2 2 . (6) I We assume frequence ω 0 for the strongest optical absorption line of the atom. Thus C = m ω 2 0 . Uncoupled energy is 2 × 1 2 0 . Notes

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I Coulomb interaction energy of the two oscillators becomes H 1 = e 2 R + e 2 R + x 1 - x 2 - e 2 R + x 1 - e 2 R - x 2 . (7) I In the approximation | x 1 | , | x 2 | << R this translates to H 1 - 2 e 2 x 1 x 2 R 3 . (8) I The total Hamiltonian H 0 + H 1 can be diagonalized by the normal mode transformation x s 1 2 ( x 1 + x 2 ) ; x a 1 2 ( x 1 - x 2 ) , (9) I or x 1 = 1 2 ( x s + x a ) ; x 2 = 1 2 ( x s - x a ) . (10) Notes
I Similarly, the momenta associated with the normal modes are p 1 1 2 ( p s + p a ) ; p 2 1 2 ( p s - p a ) . (11) I H = H 0 + H 1 becomes H = 1 2

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md2011-04-note_Part_4 - As a model lets consider two...

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