Quantum Mechanics for Mathematicians

(x 1 ,x 2 ,x 3 ) = (r,θ,φ)

x 1 x 2

x 3

r

θ

φ

Figure 8.2: Spherical coordinates.

we will have

x 1 =rsinθcosφ

x 2 =rsinθsinφ

x 3 =rcosθ

Acting onf(r,φ,θ),SO(3) will leaverinvariant, only acting non-trivially on
θ,φ. It turns out that we can cut down the space of functions to something
that will only contain one copy of the representation we want in various ways.
One way to do this is to restrict our functions to the unit sphere, i.e., look at
functionsf(θ,φ). We will see that the representations we are looking for can
be found in simple trigonometric functions of these two angular variables.
We can construct our irreducible representationsρ′lby explicitly constructing
a function we will callYll(θ,φ) that will be a highest weight vector of weight
l. The weightlcondition and the highest weight condition give two differential
equations forYll(θ,φ):
L 3 Yll=lYll, L+Yll= 0

These will turn out to have a unique solution (up to scalars).