Mathematical Methods for Physicists: A concise introduction - Site Map

SOME SPECIAL GROUPS
Figure 12.2.
Some symmetry operations are equivalent. As shown in Fig. 12.2, a two-fold
inversion axis is equivalent to a mirror plane perpendicular to the axis.
There are two di€erent ways of looking at a rotation, as shown in Fig. 12.3.
According to the so-called active view, the system (the body) undergoes a rotation
through an angle , say, in the clockwise direction about the x 3 -axis. In the passive
view, this is equivalent to a rotation of the coordinate system through the same
angle but in the counterclockwise sense. The relation between the new and old
coordinates of any point in the body is the same in both cases:
x1 0 9
ˆ x 1 cos ‡ x 2 sin ; >=
x2 0 ˆx 1 sin ‡ x 2 cos ;
…12:6†
>;
x3 0 ˆ x 3 ;
where the prime quantities represent the new coordinates.
A general rotation, re¯ection, or inversion can be represented by a linear
transformation of the form
x1 0 9
ˆ 11 x 1 ‡ 12 x 2 ‡ 13 x 3 ; >=
x2 0 ˆ 21 x 1 ‡ 22 x 2 ‡ 23 x 3 ;
…12:7†
>;
x3 0 ˆ 31 x 1 ‡ 32 x 2 ‡ 33 x 3 :
Figure 12.3.
(a) Active view of rotation; (b) passive view of rotation.
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