Mathematical Methods for Physicists: A concise introduction - Site Map

SOME SPECIAL GROUPS
0 1
1 0 0
B C
~E ˆ @ 0 1 0A:
0 0 1
These four elements form an Abelian group with the group multiplication table
shown in Table 12.5. It is easy to check this group table by matrix multiplication.
Or you can check it by analyzing the operations themselves, a tedious task. This
demonstrates the power of mathematics: when the system becomes too complex
for a direct physical interpretation, the usefulness of mathematics shows.
Table 12.5.
~E ~ ~ ~
~E ~E ~ ~ ~
~ ~ ~E ~ ~
~ ~ ~ ~E ~
~ ~ ~ ~ ~E
This symmetry group is usually labeled D 2 , a dihedral group with a twofold
symmetry axis. A dihedral group D n with an n-fold symmetry axis has n axes with
an angular separation of 2=n radians and is very useful in crystallographic study.
We next consider an example of threefold symmetry axes. To this end, let us revisit
Example 12.8. Rotations of the triangle of 08, 1208, 2408, and 3608 leave the
triangle invariant. Rotation of the triangle of 08 means no rotation, the triangle is
left unchanged; this is represented by a unit matrix (the identity element). The
other two orthogonal rotational matrices can be set up easily:
0 p 1
1=2 3 =2
~A ˆ R z …1208† ˆ@
p
A;
3 =2 1=2
0 p 1
1=2 3 =2
~B ˆ R z …2408† ˆ@
p
A;
3 =2 1=2
and
~E ˆ R z …0† ˆ 1 0
:
0 1
We notice that ~C ˆ R z …3608† ˆ ~E. The set of the three elements … ~E; ~A; ~B† forms a
cyclic group C 3 with the group multiplication table shown in Table 12.6. The z-
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