Page 200 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.5 Spectra 191

does not give results that correspond to actual observations. For this reason, we
refer to this treatment as the pseudo-Zeeman effect.
When a magnetic ®eld B is applied to a hydrogen-like atom with magnetic
moment M, the resulting potential energy V is given by the classical expression
: ì B :
V ˆÿM B ˆ L B (6:86)
"
where equation (5.81) has been introduced. If the z-axis is selected to be
parallel to the vector B, then we have
V ˆ ì B BL z =" (6:87)

If we replace the z-component of the classical angular momentum in equation
^
(6.87) by its quantum-mechanical operator, then the Hamiltonian operator H B
for the hydrogen-like atom in a magnetic ®eld B becomes
ì B B
^ ^ ^
H B ˆ H ‡ L z (6:88)
"
^
where H is the Hamiltonian operator (6.14) for the atom in the absence of the
magnetic ®eld. Since the atomic orbitals ø nlm in equation (6.56) are simultan-
^
^
^ 2
eous eigenfunctions of H, L , and L z , they are also eigenfunctions of the
^
operator H B . Accordingly, we have

^ ^ ì B B ^
H B ø nlm ˆ H ‡ L z ø nlm ˆ (E n ‡ mì B B)ø nlm (6:89)
"
where E n is given by (6.48) and equation (6.15c) has been used. Thus, the
energy levels of a hydrogen-like atom in an external magnetic ®eld depend on
the quantum numbers n and m and are given by

2
Z e9 2
E nm ˆÿ ‡ mì B B, n ˆ 1, 2, ... ; m ˆ 0, 1, ... , (n ÿ 1)
2a ì n 2
(6:90)
This dependence on m is the reason why m is called the magnetic quantum
number.
The degenerate energy levels for the hydrogen atom in the absence of an
external magnetic ®eld are split by the magnetic ®eld into a series of closely
spaced levels, some of which are non-degenerate while others are still
degenerate. For example, the energy level E 3 for n ˆ 3 is nine-fold degenerate
in the absence of a magnetic ®eld. In the magnetic ®eld, this energy level is
split into ®ve levels: E 3 (triply degenerate), E 3 ‡ ì B B (doubly degenerate),
E 3 ÿ ì B B (doubly degenerate), E 3 ‡ 2ì B B (non-degenerate), and E 3 ÿ 2ì B B
(non-degenerate). Energy levels for s orbitals (l ˆ 0) are not affected by the
application of the magnetic ®eld. Energies for p orbitals (l ˆ 1) are split by the