The two-body integrals D 2 and 2 are defined in Equations 8.65 and 8.66. To evaluate
Question:
The two-body integrals D2 and χ2 are defined in Equations 8.65 and 8.66. To evaluate D2 we 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:
Step by Step Answer:
Introduction To Quantum Mechanics
ISBN: 9781107189638
3rd Edition
Authors: David J. Griffiths, Darrell F. Schroeter