A -electron ( ) orbital

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Invariance, Ch Representations of Lie algebras: spin, products and SU(3)

Problem

d

-electron (

l=2

) orbital, five-fold degenerate in free space, sits on an ion at a site of trigonal symmetry, point group

D_3

, whose only symmetry operations are proper rotations:

E

, two three-fold rotations

2C_3

(each

120^\circ

) and three two-fold rotations

3C_2

(each

180^\circ

);

|D_3| = 6

. Character table:

A_1 = (1,1,1)

A_2 = (1,1,-1)

E = (2,-1,0)

on classes

(E, 2C_3, 3C_2)

. ((a)) Evaluate

\chi_2(\alpha) = \sin[(2+\tfrac12)\alpha]/\sin(\alpha/2)

\alpha = 0^\circ, 120^\circ, 180^\circ

. ((b)) Reduce this character on

D_3

and state the splitting of the

d

level. Check the dimensions add to 5.