Quantum Mechanics for Mathematicians

for some functionFl(θ), and using the second we get

(

∂

∂θ

−lcotθ)Fl(θ) = 0

with solution
Fl(θ) =Cllsinlθ

for an arbitrary constantCll. Finally

Yll(θ,φ) =Clleilφsinlθ

This is a function on the sphere, which is also a highest weight vector in a
2 l+ 1 dimensional irreducible representation ofSO(3). Repeatedly applying the
lowering operatorL−gives vectors spanning the rest of the weight spaces, the
functions

Ylm(θ,φ) =Clm(L−)l−mYll(θ,φ)

=Clm

(

e−iφ

(

−

∂

∂θ

+icotθ

∂

∂φ

))l−m

eilφsinlθ

form=l,l− 1 ,l− 2 ···,−l+ 1,−l
The functionsYlm(θ,φ) are called “spherical harmonics”, and they span
the space of complex functions on the sphere in much the same way that the
einθspan the space of complex-valued functions on the circle. Unlike the case
of polynomials onC^2 , for functions on the sphere, one gets finite numbers
by integrating such functions over the sphere. So an inner product on these
representations for which they are unitary can be defined by simply setting

〈f,g〉=

∫

S^2

fgsinθdθdφ=

∫ 2 π

φ=0

∫π

θ=0

f(θ,φ)g(θ,φ) sinθdθdφ (8.3)

We will not try and show this here, but for the allowable values ofl,mthe
Ylm(θ,φ) are mutually orthogonal with respect to this inner product.
One can derive various general formulas for theYlm(θ,φ) in terms of Leg-
endre polynomials, but here we’ll just compute the first few examples, with
the proper constants that give them norm 1 with respect to the chosen inner
product.

• For thel= 0 representation

Y 00 (θ,φ) =

√

1

4 π

• For thel= 1 representation

Y 11 =−

√

3

8 π

eiφsinθ, Y 10 =

√

3

4 π

cosθ, Y 1 −^1 =

√

3

8 π

e−iφsinθ

(one can easily see that these have the correct eigenvalues forρ′(L 3 ) =

−i∂φ∂).