Quantum Mechanics for Mathematicians

• Forn= 3,s=^32

1

√

6

z 13 ,

1

√

2

z 12 z 2 ,

1

√

2

z 1 z 22 ,

1

√

6

z 23

8.3 Representations ofSO(3)and spherical har-

monics

We would like to now use the classification and construction of representations of
SU(2) to study the representations of the closely related groupSO(3). For any
representation (ρ,V) ofSO(3), we can use the double covering homomorphism
Φ :SU(2)→SO(3) to get a representation

π=ρ◦Φ

ofSU(2). It can be shown that ifρis irreducible,πwill be too, so we must
haveπ=ρ◦Φ =πn, one of the irreducible representations ofSU(2) found in
the last section. Using the fact that Φ(− 1 ) = 1 , we see that

πn(− 1 ) =ρ◦Φ(− 1 ) = 1

From knowing that the weights ofπnare−n,−n+ 2,···,n− 2 ,n, we know that

πn(− 1 ) =πn

(

eiπ 0

0 e−iπ

)

=









einπ 0 ··· 0

0 ei(n−2)π ··· 0

··· ···

0 0 ··· e−inπ







=^1

which will only be true forneven, not fornodd. Since the Lie algebra ofSO(3)
is isomorphic to the Lie algebra ofSU(2), the same Lie algebra argument using
raising and lowering operators as in the last section also applies. The irreducible
representations ofSO(3) will be (ρl,V=C^2 l+1) forl= 0, 1 , 2 ,···, of dimension
2 l+ 1 and satisfying
ρl◦Φ =π 2 l
Just like in the case ofSU(2), we can explicitly construct these representa-
tions using functions on a space with anSO(3) action. The obvious space to
choose isR^3 , withSO(3) matrices acting onx∈R^3 as column vectors. The
induced representation is as usual

(ρ(g)f)(x) =f(g−^1 x)

and by the same argument as in theSU(2) case,g∈SO(3) acts on the coordi-
nates (a basis of the dualR^3 ) by

g·





x 1

x 2

x 3



=gT





x 1

x 2

x 3



