Page 178 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.3 The radial equation 169

This is the energy required to remove the electron from the ground state of a
hydrogen atom to a state of zero kinetic energy at in®nity and is also known as
the ionization potential of the hydrogen atom.

Determination of the eigenfunctions
Equation (6.47) may be used to obtain the ground state (n ˆ 1, l ˆ 0) eigen-
^
function S 10 (r). Introducing the de®nition of A n in equation (6.26a), we have
d r
^
A 1 S 10 ˆÿ r ‡ S 10 ˆ 0
dr 2
or
dS 10 S 10
ˆÿ
dr 2
from which it follows that
e
S 10 ˆ ce ÿr=2 ˆ 2 ÿ1=2 ÿr=2
where the constant c of integration was evaluated by applying equations (6.25),
(A.26), and (A.28).
The series of eigenfunctions S 20 , S 30 , ... are readily obtained from equations
(6.46) and (6.26b) with ë ˆ n, l ˆ 0
r
^ d
B n S nÿ1,0 ˆ r ÿ ‡ n S nÿ1,0 ˆ nS n0
dr 2
Thus, S 20 is

1 d r ÿ1=2 ÿr=2
S 20 ˆ r ÿ ‡ 2 2 e
2 dr 2
1
ˆ p (2 ÿ r)e ÿr=2

2 2
and S 30 is

1 d r 1 ÿr=2
S 30 ˆ r ÿ ‡ 3 p  (2 ÿ r)e
3 dr 2 2 2
1
2
ˆ p (6 ÿ 6r ‡ r )e ÿr=2

6 2
and so forth ad in®nitum. Each eigenfunction is normalized.
The eigenfunctions for l . 0 are determined in a similar manner. A general
formula for the eigenfunction S l‡1,l , which is the starting function for evaluat-
ing the series S nl with ®xed l, is obtained from equations (6.44) and (6.26a)
with l ˆ n ˆ l ‡ 1