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Physics 3220 – Quantum Mechanics 1 – Fall 2008
Problem Set #11
Due Wednesday, Nov 19 at 2pm
Problem
11. 1
Modeling molecules: a quantum rigid rotator
[15 pts]
Simple molecules can be modeled as two particles of mass m, attached to the ends of a massless rod
of total length a. The system is free to rotate in 3D.
A) Show that, classically, the total angular momentum of the rigid object described above, rotating
about a fixed axis
through its centerofmass
, is independent of the choice of origin.
Hence, the origin in this problem can be chosen to be on the axis without loss of generality.
B) The energy of this system is JUST rotational kinetic energy. Express the classical energy in terms
of the angular momentum of the system, and thus correspondingly deduce the quantum Hamiltonian.
C) Show that the allowed energies of the quantum system are
E
n
=
η
2
ν
(
+
129
μα
2
,
= 0,1,2,Κ
D) What are the normalized eigenfunctions for this system? What is the degeneracy of energy
level n?
Problem
11. 2
Measurement of atomic angular momentum.
[10
pts]
Individual atoms often have a total angular momentum of ½ ħ or 1 ħ.
In most materials, the expectation values of the angular momenta of individual
atoms point in random directions, so the net angular momentum is zero. We will
see in lecture that the angular momentum of an atom is parallel to the magnetic
moment of the atom. By applying a strong magnetic field to a magnetic material,
one can force alignment of the magnetic moments, and hence alignment of the
angular momenta.
Imagine that you have a cylinder of unmagnetized magnetic material (such as iron) suspended from
a thread, which is initially at rest.
If you apply a strong vertical magnetic field, all the angular
momenta of the atoms will align, and the cylinder should start rotating.
A) Why should the cylinder start rotating?
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 Fall '08
 STEVEPOLLOCK
 Mass

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