Chemistry Reference
In-Depth Information
α
|
β
In this way, the transformation of the spinor
(
|
α
|
β
)
→
(
|
) induces
a trans-
(x
y
z
)
. In the vector space this transformation
formation of the vector
(x y z)
→
is described by a matrix
O
(R)
. This matrix may easily be constructed by combining
the previous two equations. One has
x
z
=
ˆ
R xy )
y
=
xy )
O
(R)
(7.31)
where the transformation matrix is given by
⎛
⎞
1
−
b
2
)
i
−
b
2
)
2
(a
2
a
2
b
2
2
(a
2
a
2
b
2
+¯
−
−
−¯
+
−
ab
−
ab
⎝
⎠
i
2
(a
2
a
2
b
2
+
b
2
)
1
2
(a
2
a
2
b
2
+
b
2
)
O
(R)
=
−¯
−
+¯
+
−
i(ab
−
ab)
ab
i(ab
2
2
+¯
ab
−
−¯
ab)
|
a
|
−|
b
|
(7.32)
It can easily be shown that this matrix is an orthogonal transformation with determi-
nant equal to unity. Hence, it belongs to the
SO(
3
)
group. As a result, it will leave
the squared length of the vector invariant:
=
x
2
+
y
2
+
z
2
x
2
+
y
2
+
z
2
(7.33)
This conservation of length is the property that confirms the previous identification
of the interaction matrix elements with a 3-vector and relates it to ordinary space. In
fact, by identifying the rotation matrices in Eqs. (
7.3
) and (
7.32
), we may determine
the Cayley-Klein parameters. Two solutions with opposite signs are possible:
cos
α
in
z
sin
α
2
a
=
2
−
(7.34)
n
y
sin
α
in
x
sin
α
2
b
=
−
2
−
or
cos
α
in
z
sin
α
2
a
=−
2
−
(7.35)
−
n
y
sin
α
in
x
sin
α
2
=−
2
−
b
As this equation shows, the mapping
(R)
is not an isomorphism but a homomor-
phism (see Fig.
7.3
). Indeed, the elements of the matrix
O
are bilinear in the
a
and
b
parameters; hence, an overall sign change of the two Cayley-Klein parameters
will give the same rotational matrix. The mapping between
SU(
2
)
and
SO(
3
)
is a
two-to-one mapping. Each element of the rotation group in 3D space is the image
of two elements in
SU(
2
)
. For this reason,
SU(
2
)
is also called a covering group
of
SO(
3
)
. The unit element in
SO(
3
)
is the image of the identity matrix in
SU(
2
)
and minus the identity matrix. This homomorphism also appears when we check
the parameter list of Eq. (
7.4
), which leaves the rotation matrix of the vector un-
changed. The overall sign change of the rotation angle and the directional cosines to
O
