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InÄ   p^O v,Vrtu,<^-   ^rf+AF^+A2^-^ (12-7) I     . _     "        \However, we have already recognized that
The variable in Eq. (12-7) is A., and each power of A is linearly independent of all other powers of A. As indicated in Section 3-4D, this means that Eq. (12-7) can be satisfied for all values of A. only if it is satisfied for each power of k separately. Collecting terms having the zeroth power of k gives
(12-8)
and Eq. (12-8) is simply a restatement of Eq. (12-1). Collecting terms from Eq. (12-7) containing a. to the first power we obtain    , .    .  \A        A      ,  H\ j 1«)
This equality must hold for Smy value of A, so the term in parentheses is zero. Hence, rearranging and making use of Eqs. (12-9), we have the first-order equation
WßCvtrcT nie,   (12-9) UOoik) _
1 ^ bfi
(H'-Wfl))+i + {Ho-E,)4>jl)=0 ^s (12-11) Let us multiply this from the left by f * and integrate:
Using the hermitian property of Hq, it is easy to show that the third and fourth terms
1-
cancel, leaving
0
(12-13a) vi;