Page 170 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.3 The radial equation 161
^
Hø(r, è, j) ˆ Eø(r, è, j)
with

" 2 @ @ 1
^ 2 ^ 2
H ˆÿ r ‡ L ‡ V(r) (6:14)
2
2ìr @r @r 2ìr 2
^ 2
The operator L in equation (5.32) commutes with the Hamiltonian operator
^
^ 2
H in (6.14) because L commutes with itself and does not involve the variable
^
^
r. Likewise, the operator L z in equation (5.31c) commutes with H because it
^ 2
commutes with L as shown in (5.15a) and also does not involve the variable r.
Thus, we have
^
^
^ ^
^ ^
2
2
[H, L ] ˆ 0, [H, L z ] ˆ 0, [L , L z ] ˆ 0
^
^ ^ 2
and the operators H, L , and L z have simultaneous eigenfunctions,
^
Hø(r, è, j) ˆ Eø(r, è, j) (6:15a)
^ 2 2 l ˆ 0, 1, 2, ... (6:15b)
L ø(r, è, j) ˆ l(l ‡ 1)" ø(r, è, j),
^ m ˆÿl, ÿl ‡ 1, ... , l ÿ 1, l (6:15c)
L z ø(r, è, j) ˆ m"ø(r, è, j),
^
^ 2
The simultaneous eigenfunctions of L and L z are the spherical harmonics
^
^ 2
Y lm (è, j) given by equations (5.50) and (5.59). Since neither L nor L z involve
the variable r, any speci®c spherical harmonic may be multiplied by an
arbitrary function of r and the result is still an eigenfunction. Thus, we may
write ø(r, è, j)as
ø(r, è, j) ˆ R(r)Y lm (è, j) (6:16)
Substitution of equations (6.13), (6.14), (6.15b), and (6.16) into (6.15a) gives
^
H l R(r) ˆ ER(r) (6:17)
where
" 2 d d Ze9 2
^ 2
H l ˆÿ r ÿ l(l ‡ 1) ÿ (6:18)
2ìr 2 dr dr r
and where the common factor Y lm (è, j) has been divided out.
6.3 The radial equation

Our next task is to solve the radial equation (6.17) to obtain the radial function
R(r) and the energy E. The many solutions of the differential equation (6.17)
depend not only on the value of l, but also on the value of E. Therefore, the
2
solutions are designated as R El (r). Since the potential energy ÿZe9 =r is
always negative, we are interested in solutions with negative total energy, i.e.,
where E < 0. It is customary to require that the functions R El (r) be normal-