The two-body integrals D 2 and 2 are defined in Equations 8.65 and 8.66. To evaluate

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The two-body integrals D2 and χ2 are defined in Equations 8.65 and 8.66. To evaluate Dwe write

where θ2 is the angle between R and r2 (Figure 8.8), and

(a) Consider first the integral over r1. Align the z axis with r2 (which is a constant vector for the purposes of this first integral) so that

Do the angular integration first and show that

(b) Plug your result from part (a) back into the relation for D2, and show that

Again, do the angular integration first.
Comment: The integral χ2 can also be evaluated in closed form, but the procedure is rather involved. We will simply quote the result,

where ϒ = 0.5772 ... is Euler’s constant, Ei(x) is the exponential integral (8.71)

and Î is obtained from I by switching the sign of R:

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Introduction To Quantum Mechanics

ISBN: 9781107189638

3rd Edition

Authors: David J. Griffiths, Darrell F. Schroeter

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