One can show that (see Eyring, Walter, and Kimball, p. 369) where r is the larger of
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where r is the larger of r1 and r2. Substitute this expansion into (9.52). Then multiply the right side by
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Which from (5.101) equals 1. Use the orthonormality of the spherical harmonics [Eq. (7.27)] to evaluate the angular integrals in terms of Kronecker deltas. Perform the sums to show that
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Next, integrate first over r1 and write the r1 integral as the sum of integrals from 0 to r2 and from r2 to . In the range 0 ( r1 ( r2, we have r> = r2; in the range r2 ( r1 ( (, we have r> = r1. Use indefinite integrals in the Appendix to do the r1 integrals to obtain r2 integrals, which are evaluated using an Appendix integral. Show that the result is (9.53).
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Substitution of 9123 into 952 and multiplication by gives Use of the orthonormality of the spheric...View the full answer
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