A hypothetical ( -type) 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

A hypothetical

l = 5

(

h

-type) electron orbital, eleven-fold degenerate in free space, sits on an ion at a site of cubic symmetry, point group

O

. The classes of

O

, with size and rotation angle in brackets, are

E

(1,

0^\circ

8C_3

(8,

120^\circ

3C_2

(3,

180^\circ

6C_4

(6,

90^\circ

) and

6C_2'

(6,

180^\circ

); the order is

|O| = 24

. Irreducible characters:

A_1 = (1,1,1,1,1)

A_2 = (1,1,1,-1,-1)

E = (2,-1,2,0,0)

T_1 = (3,0,-1,1,-1)

T_2 = (3,0,-1,-1,1)

. ((a)) Using

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

, compute

\chi_5

at each of the five class angles. ((b)) Reduce this character using

a_i = \tfrac1{|G|}\sum_{\text{classes}} (\text{size})\,\chi(\text{class})\,\chi_i(\text{class})

, and state the splitting of the

l=5

level in this cubic field. Check the dimensions add to 11.