Physical Chemistry Third Edition

F Some Mathematics Used in Quantum Mechanics 1281

This equation is valid only for non-negative values of the integerm, but Eq. (F-47)

contains only the square ofm, so that we can replace a negative value ofmby|m|when

using this equation. This equation leads to theΘfunctions discussed in Chapter 17.

The radial functionRobeys Eq. (17.3-5), the associated Laguerre equation:

d^2 R

dρ^2

+

2

ρ

dR

dr

−

R

4

+

βR

ρ

−

l(l+1)R

ρ^2

 0 (F-50)

The solution to this equation can be written in the form

R(ρ)G(ρ)e−ρ/^2 (F-51)

whereG(ρ) is a polynomial related to the associated Laguerre functionsLkj(ρ):

G(ρ)NnlρlL^2 nl++ 11 (ρ) (F-52)

whereNnlis a normalizing factor, given by

Nnl

[(

2 Z

na

) 3

(n−l−1)!

2 n[(n+l)!]^3

] 1 / 2

(F-53)

The associated Laguerre functions are

Lsu(ρ)

ds

dρs

Lu(ρ) (F-54)

whereLuis the Laguerre polynomial

Lu(ρ)eρ

du

dρu

(ρue−ρ) (F-55)

These equations lead to the radial factors discussed in Chapter 17.