- 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.