Page 185 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

176 The hydrogen atom

magnetic quantum number m determines the z-component of the angular
momentum. We have found that the allowed values of n, l, and m are
m ˆ 0, 1, 2, ...

l ˆjmj, jmj‡ 1, jmj‡ 2, ...
n ˆ l ‡ 1, l ‡ 2, l ‡ 3, ...

This set of relationships may be inverted to give
n ˆ 1, 2, 3, ...

l ˆ 0, 1, 2, ... , n ÿ 1
m ˆÿl, ÿl ‡ 1, ... , ÿ1, 0, 1, ... , l ÿ 1, l

These eigenfunctions form an orthonormal set, so that
hn9l9m9jnlmiˆ ä nn9 ä ll9 ä mm9
The energy levels of the hydrogen-like atom depend only on the principal
quantum number n and are given by equation (6.48), with a ì replaced by a 0 ,as
2
Z e9 2
E n ˆÿ , n ˆ 1, 2, 3, ... (6:57)
2a 0 n 2
To ®nd the degeneracy g n of E n , we note that for a speci®c value of n there are
n different values of l. For each value of l, there are (2l ‡ 1) different values of
m, giving (2l ‡ 1) eigenfunctions. Thus, the number of wave functions corre-
sponding to n is given by
nÿ1 nÿ1 nÿ1
X X X
g n ˆ (2l ‡ 1) ˆ 2 l ‡ 1
lˆ0 lˆ0 lˆ0
The ®rst summation on the right-hand side is the sum of integers from 0 to
(n ÿ 1) and is equal to n(n ÿ 1)=2(n terms multiplied by the average value of
each term). The second summation on the right-hand side has n terms, each
equal to unity. Thus, we obtain
g n ˆ n(n ÿ 1) ‡ n ˆ n 2
2
showing that each energy level is n -fold degenerate. The ground-state energy
level E 1 is non-degenerate.
The wave functions jnlmi for the hydrogen-like atom are often called atomic
orbitals. It is customary to indicate the values 0, 1, 2, 3, 4, 5, 6, 7, ... of the
azimuthal quantum number l by the letters s, p, d, f, g, h, i, k, ... , respectively.
Thus, the ground-state wave function j100i is called the 1s atomic orbital,
j200i is called the 2s orbital, j210i, j211i, and |21 ÿ1l are called 2p orbitals,
and so forth. The ®rst four letters, standing for sharp, principal, diffuse, and