Separate the time-independent Schrodinger equation (3.22) in spherical coordinates

Chapter 13, Problem 18

(choose chapter or problem)

Separate the time-independent Schrodinger equation (3.22) in spherical coordinates assuming that V = V (r) is independent of and . (If V depends only on r, then we are dealing with central forces, for example, electrostatic or gravitational forces.) Hints: You may find it helpful to replace the mass m in the Schrodinger equation by M when you are working in spherical coordinates to avoid confusion with the letter m in the spherical harmonics (7.10). Follow the separation of (7.1) but with the extra term [V (r) E]. Show that the , solutions are spherical harmonics as in (7.10) and 16. Show that the r equation with k = l(l + 1) is [compare (7.6)] 1 R d dr r2 dR dr 2Mr2 h2 [V (r) E) = l(l + 1).