Page 177 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

168 The hydrogen atom

(6.43) and b 2 ël in equation (6.45), being squares of real numbers, must be
positive. Consequently, the factor (ë ÿ l ÿ 1) must be positive, so that
ë > (l ‡ 1).
We now select some appropriately large value î of the parameter ë in
equation (6.44) and continually apply the lowering operator to both sides of the

equation in the same manner as in the l ˆ 0 case. Eventually we obtain S îÿk,l
such that (l ‡ 1) < (î ÿ k) , (l ‡ 2). The next step in the sequence would
give S îÿkÿ1,l or S ël with ë ˆ (î ÿ k ÿ 1) , (l ‡ 1), which is not allowed, so
that the sequence must be terminated according to
^
A îÿk S îÿk,l ˆ a îÿk,l S îÿkÿ1,l
1=2
î ÿ k ÿ 1
ˆ (î ÿ k ‡ l)(î ÿ k ÿ l ÿ 1) S îÿkÿ1,l
î ÿ k
ˆ 0
for some value of k. Thus, î must be an integer for a îÿk,l to vanish. As k
increases during the sequence, the constant a îÿk,l vanishes when
k ˆ (î ÿ l ÿ 1) or (î ÿ k) ˆ (l ‡ 1). The minimum value of ë is then l ‡ 1.
Combining the conclusions of both cases, we see that the minimum value of
ë is l ‡ 1 for l ˆ 0, 1, 2, ... Beginning with the value ë ˆ l ‡ 1, we can apply
equation (6.46) to yield an in®nite progression of eigenfunctions S nl (r) for each
value of l (l ˆ 0, 1, 2, ...), where ë can take on only integral values,
ë ˆ n ˆ l ‡ 1, l ‡ 2, l ‡ 3, ... Since î in both cases was chosen arbitrarily
and was shown to be an integer, equation (6.46) generates all of the eigenfunc-
tions S ël (r) for each value of l. There are no eigenfunctions corresponding to
non-integral values of ë. Since ë is now shown to be an integer n, in the
remainder of this presentation we replace ë by n.
Solving equation (6.21) for the energy E and replacing ë by n, we obtain the
quantized energy levels for the hydrogen-like atom
2
2
ìZ e9 4 Z e9 2
E n ˆÿ ˆÿ , n ˆ 1, 2, 3, ... (6:48)
2 2
2" n 2a ì n 2
These energy levels agree with the values obtained in the earlier Bohr theory.
Electronic energies are often expressed in the unit electron volt (eV). An
electron volt is de®ned as the kinetic energy of an electron accelerated through
a potential difference of 1 volt. Thus, we have
1eV ˆ (1:602 177 3 10 ÿ19 C) 3 (1:000 000 V) ˆ 1:602 177 3 10 ÿ19 J
The ground-state energy E 1 of a hydrogen atom (Z ˆ 1) as given by equation
(6.48) is
E 1 ˆÿ2:178 68 3 10 ÿ18 J ˆÿ13:598 eV