f
R 3 ¼
cos α À sin α 0
sin α cos α 0
0
0
1
0
B
B
@
1
C
C
A
cos β 0 sin β
0
1
0
À sin β 0 cos β
0
B
B
@
1
C
C
A
cos γ À sin γ 0
sin γ
cos γ 0
0
0
1
0
B
B
@
1
C
C
A
¼
cos α cos β cos γ À sin α sin γ À cos α cos β sin γ À sin α cos γ cos α sin β
sin α cos β cos γ þ cos α sin γ À sin α cos β sin γ þ cos α cos γ sin α sin β
À sin β cos γ
sin β sin γ
cos β
0
B
B
@
1
C
C
A :
ð17:101Þ
This matrix corresponds to (17.87), where n ¼ 3. The angles α, β, and γ in (17.101)
are called Euler angles and their domains are usually taken as 0 α 2π, 0 β π,
0 γ 2π. The matrix (17.101) is widely used in quantum mechanics and related
fields of natural science. The matrix notation, however, differs from literature to
literature, and so care should be taken [3–5]. Using the notation of (17.87), we have
R 1 ¼
cos α À sin α 0
sin α
cos α 0
0
0
1
0
B
@
1
C
A,
ð17:102Þ
R
1
ð Þ
2 ¼
cos β 0 sin β
0
1
0
À sin β 0 cos β
0
B
@
1
C
A,
ð17:103Þ
R
2
ð Þ
3 ¼
cos γ À sin γ 0
sin γ
cos γ 0
0
0
1
0
B
@
1
C
A:
ð17:104Þ
From (17.94) we also get
e
P R 1 R
1
ð Þ
2 ¼
cos α cos β À sin α cos α sin β
sin α cos β
cos α
sin α sin β
À sin β
0
cos β
0
B
@
1
C
A:
ð17:105Þ
Corresponding to (17.97), we have
R 3 ¼ e
PR
2
ð Þ
3
e
P
À1 ¼ R 1 R
1
ð Þ
2 R
2
ð Þ
3
R
1
ð Þ
2
h
i À1
R 1
À1
:
ð17:106Þ
Now matrix calculations are readily performed such that
17.4 Special Orthogonal Group SO(3)
673