448
8 Electric and Magnetic Phenomena
and known as the Landé g-factor for a multi-electron atom or ion in a specific (L,S,J)
energy level. Note that we could also write the first-order Zeeman perturbation
energy as W 1 (L, S, J, M J )B = μ J M J (L, S)B, with μ J M J (L, S) ≡ μ B g J (L, S)M J
representing the magnetic moment of the multi-electron atom or ion in the energy
state E(α, L, S, J, M J ).
To determine the second-order Zeeman contribution to the energy, we note that
the component of μ
(m)
z
that is proportional to
J z is diagonal in both the J and M J
quantum numbers (because it commutes with both
J z and
J 2 ), while the component
proportional to
L z , which commutes with
J z but not with
J 2 , although diagonal in
M J , is not diagonal in J . The total orbital angular momentum operator L happens,
however, to belong to a class of vector operators V whose Cartesian components
satisfy the nine commutation relations represented collectively by the equation
[
J ,
V m ] − = iε mn
V n ,
in which the subscripts m, n take on values x, y, z,and the symbol ε mn is +1 for
mn an even permutation of xyz, −1 for mn an odd permutation of xyz, and 0
otherwise. The Wigner–Eckart theorem applies precisely to this class of operators,
so that it may be utilized to compute the non-vanishing off-diagonal elements
of
L z . According to the Wigner–Eckart theorem, such a vector operator has offdiagonal elements between the J and J = J ± 1 eigenstates of
J 2 . For present
purposes, the relevant eigenstates are the coupled eigenstates |α, L, S, J, M J , and
the appropriate matrix elements for the operator
L z are
α, L, S, J − 1, M J |
L z |α, L, S, J, M J = =α, L, S, J, M J |
L z |α, L, S, J − 1, M J
= −
f (L, S, J )(J 2 − M 2
J ),
(8.5.31)
and
α, L, S, J + 1, M J |
L z |α, L, S, J, M J = =α, L, S, J, M J |
L z |α, L, S, J + 1, M J
= −
f (L, S, J + 1)[(J + 1) 2 − M 2
J ],
(8.5.32)
with f (L, S, J ) defined as
f (L, S, J ) ≡
[J 2 − (L − S) 2 ][(L + S + 1) 2 − J 2 ]
4J 2 (4J 2 − 1)
.
(8.5.33)
We may obtain the second-order perturbation energy W 2 B 2 from these results,
with W 2 given as
8 Electric and Magnetic Phenomena
and known as the Landé g-factor for a multi-electron atom or ion in a specific (L,S,J)
energy level. Note that we could also write the first-order Zeeman perturbation
energy as W 1 (L, S, J, M J )B = μ J M J (L, S)B, with μ J M J (L, S) ≡ μ B g J (L, S)M J
representing the magnetic moment of the multi-electron atom or ion in the energy
state E(α, L, S, J, M J ).
To determine the second-order Zeeman contribution to the energy, we note that
the component of μ
(m)
z
that is proportional to
J z is diagonal in both the J and M J
quantum numbers (because it commutes with both
J z and
J 2 ), while the component
proportional to
L z , which commutes with
J z but not with
J 2 , although diagonal in
M J , is not diagonal in J . The total orbital angular momentum operator L happens,
however, to belong to a class of vector operators V whose Cartesian components
satisfy the nine commutation relations represented collectively by the equation
[
J ,
V m ] − = iε mn
V n ,
in which the subscripts m, n take on values x, y, z,and the symbol ε mn is +1 for
mn an even permutation of xyz, −1 for mn an odd permutation of xyz, and 0
otherwise. The Wigner–Eckart theorem applies precisely to this class of operators,
so that it may be utilized to compute the non-vanishing off-diagonal elements
of
L z . According to the Wigner–Eckart theorem, such a vector operator has offdiagonal elements between the J and J = J ± 1 eigenstates of
J 2 . For present
purposes, the relevant eigenstates are the coupled eigenstates |α, L, S, J, M J , and
the appropriate matrix elements for the operator
L z are
α, L, S, J − 1, M J |
L z |α, L, S, J, M J = =α, L, S, J, M J |
L z |α, L, S, J − 1, M J
= −
f (L, S, J )(J 2 − M 2
J ),
(8.5.31)
and
α, L, S, J + 1, M J |
L z |α, L, S, J, M J = =α, L, S, J, M J |
L z |α, L, S, J + 1, M J
= −
f (L, S, J + 1)[(J + 1) 2 − M 2
J ],
(8.5.32)
with f (L, S, J ) defined as
f (L, S, J ) ≡
[J 2 − (L − S) 2 ][(L + S + 1) 2 − J 2 ]
4J 2 (4J 2 − 1)
.
(8.5.33)
We may obtain the second-order perturbation energy W 2 B 2 from these results,
with W 2 given as