Atomic Orbitals - Methods of Molecular Quantum Mechanics - page 42

characterizes the energy levels and the orbital quantum number char-

acterizes the square of the angular momentum, the so-called magnetic
number m, the z-component of the angular momentum.
Now, a fundamental theorem of quantum mechanics (Eyring et al.,
1944) states that, when the same function is simultaneously an eigenfunc-
tion of different operators, the corresponding operators commute in pairs;
namely:

'

2 2

½ H ; L ¼½ H ; L z ¼½ L ; L z ¼ 0 ð 3 : 43 Þ

Operators commuting with the Hamiltonian are called constants of the
motion, which means that energy, the square of the angular momentum
and the z-component of the angular momentum can all be measured with
arbitrary accuracy at the same time.
The discrete spectrum of the first few energy levels of the bound
hydrogen-like electron up to n ¼ 3 is given in Figure 3.3. It is seen that,

E < 0

3s, 3p, 3d

n = 3

g = 9

2s, 2p

n = 2

g = 4

1s
g = 1

n = 1

Figure 3.3 Schematic diagram of the energy levels of the hydrogen-like atom up to
n ¼ 3