6.3 Zeeman effect and hyperfine structure 111good quantum number andJprecesses aboutB.^27 The effect of the hy-^27 The nuclear angular momentum I
does not precess aroundB because
−μI·Bis negligible. In this regime the
interactionAI·JmakesIprecess about
the mean direction ofJ, which is par-
allel toB. Thus effectivelyIprecesses
about the axis defined byB(but not
because of−μI·B). The vector model
picture requires careful thought be-
cause of the subtle differences from the
Paschen–Back effect (Fig. 5.14). In the
quantum mechanical description this
is taken into account by considering
the relative magnitudes of the pertur-
bations: |μe·B|>|AI·J|>|μI·B|,
whereμeis the magnetic moment of
the atomic electrons in eqn 5.9.
perfine interaction can be calculated as a perturbation on the|IMIJMJ〉
eigenstates, i.e.
EZE=gJμBBMJ+〈IMIJMJ|AI·J|IMIJMJ〉 (6.32)
=gJμBBMJ+AMIMJ. (6.33)The first term is the same as eqn 5.11. In the second term,I·J=
IxJx+IyJy+IzJzand thex-andy-components average to zero in the
precession about the field along thez-direction.^28
(^28) This can be shown rigorously using
the ladder operators
I+≡Ix+iIy,
I−≡Ix−iIy,
and similarly forJ+andJ−.These
ladder operators change the magnetic
quantum numbers, e.g.
I+|IMI〉∝|IMI+1〉.
Since
IxJx+IyJy=^1
2
(I+J−+I−J+),
the expectation value of this part ofI·J
is zero (for states of givenMJandMI
as in eqn 6.32).
An example of the energy levels in a strong field is shown in Fig. 6.10
for the hydrogen ground state. The two energy levels withMJ=± 1 / 2
are both split into sub-levels withMI=± 1 /2 by the hyperfine inter-
action; eqn 6.33 shows that these sub-levels have a separation ofA/ 2
(independent of the field strength).
6.3.3 Intermediate field strength
In Fig. 6.10 the low- and high-field energy levels follow the rule that two
states never cross if they have the same value ofM, where at low fields
M=MFandathighfieldsM=MI+MJ. This implies that
MJ MI
F=1,MF=0 → +1/ 2 , − 1 / 2 ,
F=0,MF=0 →− 1 / 2 , +1/ 2.
This rule can be justified by showing that the operatorIz+Jzcom-
mutes with all the interactions and it allows unambiguous connection
of states even in more complex cases.^29 For the simple case of hydrogen
(^29) At low fields,Iz+Jz≡Fz,which
clearly commutes with the interaction
in eqn 6.29. At high fields the relevant
interactions are proportional toJzand
IxJx+IyJy+IzJz,bothofwhichcom-
mute withIz+Jz.
the energy levels can be calculated at all fields by simple perturbation
theory, as shown below.
Example 6.3 The Zeeman effect on the hyperfine structure of hydrogen
for all field strengths
Figure 6.10 shows the energy levels for all field strengths. The Zeeman
energies of theM =±1 states are±μBBfor all fields because their
wavefunctions are not mixed (gF = 1 from eqn 6.30). TheMF =0
states have no first-order shift but the magnetic field mixes these two
states in theF= 0 and 1 hyperfine levels; the matrix element between
them is
−〈F=1,MF=0|μ·B|F=0,MF=0〉=ζμBB.
Such (off-diagonal) matrix elements can be evaluated by angular momen-
tum theory, but in this simple case we can get by without using Clebsch–
Gordan coefficients (leavingζas an undetermined constant for the time
being). The Hamiltonian for the two-level system is
H=
(
A/ 2 ζμBB
ζμBB −A/ 2