Introduction to Quantum Mechanics, Spring 2015
Problem Set 10
Due Tuesday, February 17
Problem 1: Consider the 2d isotropic oscillator. There are two different U (1) =
SO(2) group actions on this problem, with corresponding operators:
• The rotation action on position space, with a simultaneous rotation action
on momentum space. The operator here will be the angular momentum
operator Q1 P2 − Q2 P1 given in the notes.
• Simultaneous rotations in the q1 , p1 and q2 , p2 planes. The operator here
will be the Hamiltonian.
For each case, the state space F2 will be a representation of the group U (1) =
SO(2). For each energy eigenspace, which irreducible representations (weights)
occur?
Problem 2: Consider the d = 3 isotropic harmonic oscillator. The group SO(3)
acts on the system by rotations of the position space R3 , and the corresponding
Lie algebra action on the state space F3 is given in the notes as the operators
Ul01 , Ul02 , Ul03
Exponentiating to get an SO(3) representation by operators U (g), show that
acting by such operators on the aj by conjugation
aj → U (g)aj U (g)−1
one gets the same action as the standard action of a rotation on coordinates on
R3 .
Problem 3: In the same context as problem 2, consider the states Hn with total
number eigenvalue n. These are irreducible representations of SU (3). They are
also representations of the SO(3) action of problem 2. For n = 0, 1, 2, find which
irreducible representations of SO(3) occur in Hn , and explicitly construct these
representations.
For the case of general n, the rule for which irreducibles of SO(3) will occur is
stated in the book and a reference is given. For an extra credit challenge, see if
you can derive the rule (it’s not easy...).
Problem 4: Consider the fermionic oscillator, for d = 3 degrees of freedom,
with Hamiltonian
3
H=
1X
(aF †j aF j − aF j aF †j )
2 j=1
1
• Use fermionic annihilation and creation operators to construct a representation of the Lie algebra u(3) = u(1) + su(3) on the fermionic state space
HF . Which irreducible representations of su(3) occur in this state space?
Picking a basis Xj of u(3) and bases for each irreducible representation
you find, what are the representation matrices (for each Xj ) for each such
irreducible representation?
• Consider the subgroup SO(3) ⊂ U (3) of real orthogonal matrices, and the
Lie algebra representation of so(3) on HF one gets by restriction of the
above representation. Which irreducible representations of SO(3) occur
in the state space?
2