Mathematical Methods for Physics and Engineering - Matematica.NET

29.2 CHOOSING AN APPROPRIATE FORMALISMP1P3P3R322Q11 2RQ R(a) (b) (c)QFigure 29.2 Diagram (a) shows the definition of the basis vector, (b) showsthe effect of applying a clockwise rotation of 2π/3 and (c) shows the effect ofapplying a reflection in the mirror axis through Q.from which it can be verified that D(C)D(B) =D(E). ◭Whilst a representation obtained in this way necessarily has the same dimensionas the order of the group it represents, there are, in general, square matrices ofboth smaller and larger dimensions that can be used to represent the group,though their existence may be less obvious.One possibility that arises when the group elements are symmetry operationson an object whose position and orientation can be referred to a spacecoordinate system is called the natural representation. In it the representativematrices D(X) describe, in terms of a fixed coordinate system, what happensto a coordinate system that moves with the object when X is applied. Thereis usually some redundancy of the coordinates used in this type of representation,since interparticle distances are fixed and fewer than 3N coordinates,where N is the number of identical particles, are needed to specify uniquelythe object’s position and orientation. Subsection 29.11.1 gives an example thatillustrates both the advantages and disadvantages of the natural representation.We continue here with an example of a natural representation that has no suchredundancy.◮Use the fact that the group considered in the previous worked example is isomorphic tothe group of two-dimensional symmetry operations on an equilateral triangle to generate athree-dimensional representation of the group.Label the triangle’s corners as 1, 2, 3 and three fixed points in space as P, Q, R, so thatinitially corner 1 lies at point P, 2 lies at point Q, and 3 at point R. We take P, Q, R asthe components of the basis vector.In figure 29.2, (a) shows the initial configuration and also, formally, the result of applyingthe identity I to the triangle; it is therefore described by the basis vector, (P Q R) T .Diagram (b) shows the the effect of a clockwise rotation by 2π/3, corresponding toelement A in the previous example; the new column matrix is (Q R P) T .Diagram (c) shows the effect of a typical mirror reflection – the one that leaves thecorner at point Q unchanged (element D in table 28.8 and the previous example); the newcolumn matrix is now (R Q P) T .In similar fashion it can be concluded that the column matrix corresponding to elementB, rotationby4π/3, is (R P Q) T , and that the other two reflections C and E result in1081