Introduction to Quantum Mechanics, Fall 2014
Problem Set 4
Due Tuesday, October 21
Problem 1: Using the definition
Z
2
2
1
< f, g >= 2
f (z1 , z2 )g(z1 , z2 )e−(|z1 | +|z2 | ) dx1 dy1 dx2 dy2
π C2
for an inner product on polynomials on homogeneous polynomials on C2
• Show that the representation π on such polynomials given in class (induced
from the SU (2) representation on C2 ) is a unitary representation with
respect to this inner product.
• Show that the
zj zk
√1 2
j!k!
are orthonormal with respect to this inner product (break up the integrals
into integrals over the two complex planes, use polar coordinates).
• Show that the differential operator π 0 (S3 ) is self-adjoint. Show that π 0 (S− )
and π 0 (S+ ) are adjoints of each other.
Problem 2: Using the formulas for the Y1m (θ, φ) and the inner product given
in the notes, show that
• The Y11 , Y10 , Y1−1 are orthonormal.
• Y11 is a highest weight vector.
• Y10 and Y1−1 can be found by repeatedly applying L− to a highest weight
vector.
Problem 3: Recall that the Casimir operator L2 of so(3) is the operator that
in any representation ρ is given by
L2 = L21 + L22 + L23
Show that this operator commutes with the ρ0 (X) for all X ∈ so(3). Use
this to show that L2 has the same eigenvalue on all vectors in an irreducible
representation of so(3).
Problem 4: For the case of the SU (2) representation π on polynomials on C2
given in the notes, find the Casimir operator
L2 = π 0 (S1 )π 0 (S1 ) + π 0 (S2 )π 0 (S2 ) + π 0 (S3 )π 0 (S3 )
as a explicit differential operator. Show that homogeneous polynomials are
eigenfunctions, and calculate the eigenvalues.
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