HW11 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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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 3-D. A) Show that, classically, the total angular momentum of the rigid object described above, rotating about a fixed axis through its center-of-mass , 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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HW11 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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