Quantum Mechanics for Mathematicians

• Show that in the quantized theory the angular momentum operators and
  theSO(3) Casimir operator satisfy

[Lj,H] = 0, [L^2 ,H] = 0

• Show that for a fixed energyE, the subspaceHE⊂Hof states of energy
  Ewill be a Lie algebra representation ofSO(3). Decomposing into irre-
  ducibles, this can be characterized by the various spin valueslthat occur,
  together with their multiplicity.


• Show that if a state of energyElies in a spin-lirreducible representation
  ofSO(3) at timet= 0, it will remain in a spin-lirreducible representation
  at later times.

Problem 2:
If
w=

1

m

(l×p) +e^2

q

|q|

is the Lenz vector, show that its components satisfy

{wj,h}= 0

for the Hydrogen atom Hamiltonianh.

Problem 3:
For the one dimensional quantum harmonic oscillator:

• Compute the expectation values in the energy eigenstate|n〉of the follow-
  ing operators
  Q, P, Q^2 , P^2
  and
  Q^4


• Use these to find the standard deviations in the statistical distributions of
  observed values ofqandpin these states. These are

∆Q=

√

〈n|Q^2 |n〉−〈n|Q|n〉^2 , ∆P=

√

〈n|P^2 |n〉−〈n|P|n〉^2

• For two energy eigenstates|n〉and|n′〉, find

〈n′|Q|n〉and〈n′|P|n〉

Problem 4:
Show that the functions 1,z,z,zzof section 22.4 give a basis of a Lie algebra
(with Lie bracket the Poisson bracket of that section). Show that this is a semi-
direct product Lie algebra, and that the harmonic oscillator state space gives a
unitary representation of this Lie algebra.