Quantum Mechanics for Mathematicians

• 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?