Chemistry Reference
In-Depth Information
Using Eq. (2) one can write
ρ
αβ
χ
β
|
σ
|
χ
α
σ
≡
σ
=
ˆ
(227)
αβ
Directing the axis
z
along the total field
A
, one has
σ
≡
ˆ
(
σ
−
+
σ
)
e
x
+
i
(
σ
−
−
σ
)
e
y
+
σ
z
e
z
+
+
=
(
|
χ
+
χ
−
| + |
χ
−
χ
+
|
)
e
x
+
i
(
|
χ
+
χ
−
| − |
χ
−
χ
+
|
)
e
y
+
(
|
χ
−
χ
−
| − |
χ
+
χ
+
|
)
e
z
(228)
Then, one obtains
σ
x
=
ρ
+−
χ
−
|
σ
x
|
χ
+
+
ρ
−+
χ
+
|
σ
x
|
χ
−
=
ρ
−+
+
ρ
+−
=
2Re
ρ
−+
χ
σ
y
χ
+
χ
σ
y
χ
σ
y
=
ρ
ρ
+−
−
+
−+
+
−
=
i
(
ρ
−+
−
ρ
)
=
2Im
ρ
+−
−+
σ
z
χ
σ
z
χ
χ
+
χ
σ
z
=
ρ
ρ
++
+
+
−−
−
−
=
ρ
−−
−
ρ
++
=
1
−
2
ρ
(229)
++
Now, Eq. (223) can be transformed as:
)
θ
σ
x
=
ρ
−+
+
ρ
+−
=
(
ρ
++
−
ρ
+
iω
0
(
ρ
−+
−
ρ
)
+
R
x
−−
+−
θσ
z
+
=−
ω
0
σ
y
+
R
x
σ
y
=
i
(
ρ
−+
−
ρ
)
=
i
(
iω
0
ρ
−+
+
iω
0
ρ
)
+
R
y
+−
+−
=−
ω
0
σ
x
+
R
y
)
θ
σ
z
=
(
ρ
+−
+
ρ
+
R
z
−+
θσ
x
+
=
R
z
(230)
or
θ
e
y
σ
=
˙
[
σ
×
(
ω
0
+
)]
+
R
,
ω
0
=
ω
0
e
z
,
=
(231)
This is a Larmor equation for the classical vector
σ
in the frame rotating with
frequency
due to the time dependence of the spin Hamiltonian. The relaxation
