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B. First-Order Approximations (1) First-Order Time-Dependent Perturbation Theory23 Substitution of Eq. (4) in Eq. (I), also utilizing Eq. , which is any one of y;, and integration over all space, leads t o ihbfe-iw;t/fi= > , e - i W A h 3 1y ? v y j (115) 38 HENRY AROESTE Defining o,,, = (W,, - W,)/ti and vr, = j-y;vy5 (116) (117) (€7 Eq. (115) may be transcribed more compactly: a,, = (iE)-lzV,+z, i (118) eimf'dt Up to and including Eq. (118)it was not really necessary for us to regard V as small, and thus Eq.
3) Strong Coupling The need for another form of approximate solution for Eq. (178) appears when k , sk,, where often all U , , are small enough to overlook except U,, and Unzrwhich may be rather large. I n other words, there is strong coupling between the I and n stages. Equation (178) then reduces to the two coupled equations: + (V2+ k: (V2 - U,,)F, - UtnFn (184a) Unn)Fn= U,nF, (1Mb) 52 HENRY AROESTE Equations (184)may be uncoupled in the case of exact resonance when k , = k , = k. For such collisions it is plausible that U,,= U,,and that U,, = U,,,whence Eq.
50 HENRY AROESTE in Eq. 8 (177) Having displayed Eqs. (176) and (177) in full subscript language for completeness, before we carry on it would be well to suppress all the “ab” subscripts and abbreviate n’f and IZP as IZ’ and respectively. Equation (176) would then appear as follows: (V’ + k:)Fn(r) = 2 UnlnFn? n‘ (178) In Eq. (178) Born’s approximation amounts to taking all the terms in the series as zero with the exception of the one which overlaps with the initial-state function ylm,hereafter designated as y,.
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