A square-based pyramidal antenna mount has point-group symmetry : a four-fold axis…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Representations, Schur's lemma and characters

Problem

A square-based pyramidal antenna mount has point-group symmetry C4vC_{4v}: a four-fold axis zz through the apex, two vertical mirror planes σv\sigma_v through the midpoints of opposite base edges, and two diagonal mirror planes σd\sigma_d through opposite corners. Its character table (classes EE, 2C42C_4, C2C_2, 2σv2\sigma_v, 2σd2\sigma_d) is A1=(1,1,1,1,1)A_1=(1,1,1,1,1), A2=(1,1,1,−1,−1)A_2=(1,1,1,-1,-1), B1=(1,−1,1,1,−1)B_1=(1,-1,1,1,-1), B2=(1,−1,1,−1,1)B_2=(1,-1,1,-1,1), E=(2,0,−2,0,0)E=(2,0,-2,0,0). Let D(g)D(g) be the 3×33\times3 matrix that sends a displacement (x,y,z)(x,y,z) of the apex atom to its image under gg, with the axes held fixed. ((a)) Find the character of DD on each class, using the fact that a proper rotation by angle θ\theta about zz has trace 1+2cos⁡θ1+2\cos\theta and a reflection in 33 dimensions always has trace 11. ((b)) Decompose DD into irreducible representations of C4vC_{4v}.

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