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Better Approximate Solutions of the Time-Independent Schrodinger Equation In this section we will summarize some methods which are more appropriate for slow collisions. Before pursuing these, however, we will deal briefly in Section IV-C (1) with a method of general applicability to spherically symmetric potentials. We do this now because the later elaboration of the slow-collision methods in Sections IV-C (2) to IV-C (4)will show that these are generally feasible only when approximations are made to reduce them at least partially to the methodology of Section IV-C (1).

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. (178) yields with some maneuvering the following set of simultaneous equations : [V2 + + Pa -V ZZ + UZ")I(F,3- F,) =0 (1W (18w (UZZ - Uzn)1(F, - F,) = 0 With the standard simplification that both U,,and U,,are spherically symmetric, the above equations are reduced to the methodology of Section IV-C (l),and the rest is detail.

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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ADVANCES IN CHEMICAL PHYSICS VOLUME 6 by I. PRIGOGINE


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