Chemistry Reference
In-Depth Information
e
2
2mc
2
A
2
H
2
¼
ð
10
:
64
Þ
with (10.63) being linear and (10.64) quadratic in A.
The vector potential A originates a magnetic field H at point r, given by
the vector product
i j k
H
x
H
y
H
z
xyz
1
2
H
r
¼
1
2
A
¼
ð
10
:
65
Þ
where the components of the field are constant. Then, the last term
in (10.63) is zero:
r
A
¼
div A
¼
0
ð
10
:
66
Þ
Next, it can be easily shown that
ð
H
r
Þr¼
H
ð
r
rÞ
ð
10
:
67
Þ
H
1
:
so that, taking into account spin, we get for
H
1
¼ðm
L
þ
g
e
b
e
m
S
Þ
H
ð
10
:
68
Þ
where g
e
2 is the intrinsic g-factor for the single electron (its correct
value depends on considerations of quantum electrodynamics),
b
e
is the
Bohr magneton, and
m
S
are the vector operators for the orbital and
spin magnetic moments. It should be noted that the magnetic moments
have a direction opposite to that of the vectors representing orbital and
spin angular momenta.
For a magnetic field H uniform along z:
m
L
and
H
x
¼
H
y
¼
0
;
H
z
¼
H
;
H
¼
kH
ð
10
:
69
Þ
where the field strength H should not be confused with the Hamiltonian
symbol. Equation 10.65 then becomes
i j k
00H
xyz
1
2
H
r
¼
1
2
1
2
H
ð
iy
þ
jx
Þ
A
¼
¼
ð
10
:
70
Þ
