Introduction to Quantum Mechanics, Fall 2014
Problem Set 8
Due Tuesday, December 16
Problem 1: Consider the classical Hamiltonian function for a particle moving
in a central potential
h=
1 2
(p + p22 + p23 ) + V (r)
2m 1
where
r2 = q12 + q22 + q32
• Show that the angular momentum functions lj satisfy
{lj , h} = 0
and note that this implies that the lj are conserved functions along classical trajectories.
• Show that in the quantized theory the angular momentum operators and
the SO(3) Casimir operator satisfy
[Lj , H] = 0, [L2 , H] = 0
• Show that for a fixed energy E, the subspace HE ⊂ H of states of energy
E will be a Lie algebra representation of SO(3). Decomposing into irreducibles, this can be characterized by the various spin values l that occur,
together with their multiplicity.
• Show that if a state of energy E lies in a spin-l irreducible representation
of SO(3) at time t = 0, it will remain in a spin-l irreducible representation
at later times.
Problem 2: If
w=
1
q
(l × p) + e2
m
|q|
is the Lenz vector, show that its components satisfy
{wj , h} = 0
for the Hydrogen atom Hamiltonian h.
Problem 3: For the one-dimensional quantum harmonic oscillator, compute
1
the expectation values in the energy eigenstate |ni of the following operators
Q, P, Q2 , P 2
and
Q4
Use these to find the standard deviations in the statistical distributions of observed values of q and p in these states. These are
p
p
∆Q = hn|Q2 |ni − hn|Q|ni2 , ∆P = hn|P 2 |ni − hn|P |ni2
For two energy eigenstates |ni and |n0 i, find
hn0 |Q|ni and hn0 |P |ni
Problem 4: Consider the harmonic oscillator in three dimensions, with the
Hamiltonian
H=
1
1
(P12 + P22 + P32 ) + mω 2 (Q21 + Q22 + Q23 )
2m
2
• Find the energy eigenvalues for this Hamiltonian.
• Consider the three lowest different possible values of the energy. Find
explicitly the energy eigenstates with these three energies, in both the
Schrödinger representation (in terms of wavefunctions on R3 ), and in the
Bargmann-Fock representation (as holomorphic functions of three variables).
• Find the angular momentum operators in both the Schrödinger representation and Bargmann-Fock representations (i.e. in terms of annihilation
and creation operators).
• Looking at how the angular momentum operators act on the energy eigenstates explicitly found above, which irreducible representations of the rotation group SO(3) occur amongst them?
2