- The groupSO(3) acts on the system by rotations of the position spaceR^3 ,
and the corresponding Lie algebra action on the state spaceF 3 is given in
section 25.4.2 as the operators
Ul′ 1 ,Ul′ 2 ,Ul′ 3
Exponentiating to get anSO(3) representation by operatorsU(g), show
that acting by such operators on theajby conjugation
aj→U(g)ajU(g)−^1
one gets the same action as the standard action of a rotation on coordinates
onR^3.
- The energy eigenspaces are the subspacesHn ⊂ Hwith total number
eigenvaluen. These are irreducible representations ofSU(3). They are
also representations of theSO(3) rotation action. Derive the rule for which
irreducibles ofSO(3) will occur inHn.
Problem 3:
Prove the relation of equation 26.16.
Problem 4:
Compute
τ〈^0 |N|^0 〉τ
as a function ofτ, for| 0 〉τthe squeezed state of equation 26.19 andNthe usual
number operator.
B.13 Chapters 27 and 28
Problem 1:
Consider the fermionic oscillator, ford= 3 degrees of freedom, with Hamil-
tonian
H=
1
2
∑^3
j=1
(aF†jaFj−aFjaF†j)
- Use fermionic annihilation and creation operators to construct a represen-
tation of the Lie algebrau(3) =u(1) +su(3) on the fermionic state space
HF. Which irreducible representations ofsu(3) occur in this state space?
Picking a basisXj ofu(3) and bases for each irreducible representation
you find, what are the representation matrices (for eachXj) for each such
irreducible representation? - Consider the subgroupSO(3)⊂U(3) of real orthogonal matrices, and the
Lie algebra representation ofso(3) onHF one gets by restriction of the
above representation. Which irreducible representations ofSO(3) occur
in the state space?