Quantum Mechanics for Mathematicians

• For thel= 2 representation one has

Y 22 =

√

15

32 π

ei^2 φsin^2 θ, Y 21 =−

√

15

8 π

eiφsinθcosθ

Y 20 =

√

5

16 π

(3 cos^2 θ−1)

Y 2 −^1 =

√

15

8 π

e−iφsinθcosθ, Y 2 −^2 =

√

15

32 π

e−i^2 φsin^2 θ

We will see in chapter 21 that these functions of the angular variables in
spherical coordinates are exactly the functions that give the angular depen-
dence of wavefunctions for the physical system of a particle in a spherically
symmetric potential. In such a case theSO(3) symmetry of the system implies
that the state space (the wavefunctions) will provide a unitary representationπ
ofSO(3), and the action of the Hamiltonian operatorHwill commute with the
action of the operatorsL 3 ,L±. As a result, all of the states in an irreducible
representation component ofπwill have the same energy. States are thus orga-
nized into “orbitals”, with singlet states called “s” orbitals (l= 0), triplet states
called “p” orbitals (l= 1), multiplicity 5 states called “d” orbitals (l= 2), etc.

8.4 The Casimir operator

For bothSU(2) andSO(3), we have found that all representations can be
constructed out of function spaces, with the Lie algebra acting as first-order
differential operators. It turns out that there is also a very interesting second-
order differential operator that comes from these Lie algebra representations,
known as the Casimir operator. For the case ofSO(3):

Definition(Casimir operator forSO(3)).The Casimir operator for the repre-
sentation ofSO(3)on functions onS^2 is the second-order differential operator

L^2 ≡L^21 +L^22 +L^23

(the symbolL^2 is not intended to mean that this is the square of an operatorL)

A straightforward calculation using the commutation relations satisfied by
theLjshows that
[L^2 ,ρ′(X)] = 0

for anyX∈so(3). Knowing this, a version of Schur’s lemma says thatL^2 will act
on an irreducible representation as a scalar (i.e., all vectors in the representation
are eigenvectors ofL^2 , with the same eigenvalue). This eigenvalue can be used
to characterize the irreducible representation.
The easiest way to compute this eigenvalue turns out to be to act withL^2 on
a highest weight vector. First one rewritesL^2 in terms of raising and lowering