1.3 Operations and Operators
9
coordinates are rotated in the opposite direction (hence α is replaced by −α), one
obtains
x
′ = x cos α + y sin α
y
′ =−x sin α + y cos α
(1.22)
Invert these equations to express x and y as a function of x ′ and y ′ :
x = x
′ cos α − y
′ sin α
y = x
′ sin α + y
′ cos α
(1.23)
The partial derivatives needed in the chain rule can now be obtained by direct derivation:
∂x
∂x ′ = cos α
∂y
∂x ′ = sin α
∂x
∂y ′ =−sin α
∂y
∂y ′ = cos α
(1.24)
Hence, the transformation of the derivatives is entirely similar to the transformation
of the x,y functions themselves:
ˆ
R
∂
∂x
∂
∂y
=
∂
∂x
∂
∂y
cos α − sin α
sin α cos α
(1.25)
In an operator formalism, we should denote this as
ˆ
R,
∂
∂x
= cos α
∂
∂x
+ sin α
∂
∂y
ˆ
R,
∂
∂y
=−sin α
∂
∂x
+ cos α
∂
∂y
(1.26)
As a further example, consider a symmetry transformation of a symmetry operator
itself. Take, for instance, a rotation, ˆ
C x
2 , corresponding to a rotation about the x-axis
of 180 ◦ and rotate this 90 ◦ counterclockwise about the z-direction by ˆ
C
z
4 . Applying
the general expression for an operator transformation yields
ˆ
C
x
2
′ = ˆ
C
z
4
ˆ
C
x
2
ˆ
C
z
4
−1
(1.27)
The result of this transformation corresponds to an equivalent twofold rotation
around the y-direction, ˆ
C
y
2 . The rotation around x is thus mapped onto a rotation