Quantum Mechanics for Mathematicians

• Show that the representationπon such polynomials given in section 8.2
  (induced from theSU(2) representation onC^2 ) is a unitary representation
  with respect to this inner product.


• Show that the monomials
  z 1 jz 2 k
  √
  j!k!
  are orthonormal with respect to this inner product (hint: break up the
  integrals into integrals over the two complex planes, use polar coordinates).


• Show that the differential operatorπ′(S 3 ) is self-adjoint. Show thatπ′(S−)
  andπ′(S+) are adjoints of each other.

Problem 2:
Using the formulas for theY 1 m(θ,φ) and the inner product of equation 8.3,
show that

• TheY 11 ,Y 10 ,Y 1 −^1 are orthonormal.


• Y 11 is a highest weight vector.


• Y 10 andY 1 −^1 can be found by repeatedly applyingL−to a highest weight
  vector.

Problem 3:
Recall that the Casimir operatorL^2 ofso(3) is the operator that in any
representationρis given by

L^2 =L^21 +L^22 +L^23

Show that this operator commutes with theρ′(X) for allX ∈so(3). Use
this to show thatL^2 has the same eigenvalue on all vectors in an irreducible
representation ofso(3).

Problem 4:
For the case of theSU(2) representationπon polynomials onC^2 given in
the notes, find the Casimir operator

L^2 =π′(S 1 )π′(S 1 ) +π′(S 2 )π′(S 2 ) +π′(S 3 )π′(S 3 )

as an explicit differential operator. Show that homogeneous polynomials are
eigenfunctions, and calculate the eigenvalues.

B.5 Chapter

Problem 1:
Consider the action ofSU(2) on the tensor productV^1 ⊗V^1 of two spin
representations. According to the Clebsch-Gordan decomposition, this breaks
up into irreducibles asV^0 ⊕V^2.