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HW11 (2)

# HW11 (2) - Physics 3220 Quantum Mechanics 1 Spring 2009...

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Physics 3220 – Quantum Mechanics 1 – Spring 2009 Problem Set #11 Due Wednesday, April 15 at 9am Problem 11.1 : Modeling molecules: the Quantum Rigid Rotor. (20 points) Simple diatomic molecules can be modeled as two particles of mass m (representing the atoms), attached to the ends of a massless rod of total length a . The system is free to rotate in three dimensions, but we will assume the center of mass is not moving. 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 rotational kinetic energy. Express the classical energy in terms of the angular momentum of the system, and correspondingly deduce the quantum Hamiltonian. c) Show that the allowed energies of the quantum system are E n = ¯ h 2 n ( n + 1) ma 2 , n = 0 , 1 , 2 , . . . . (1) 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 points) Individual atoms often have a total angular momentum of (1 / 2)¯ h or 1¯ h . In most materi-

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HW11 (2) - Physics 3220 Quantum Mechanics 1 Spring 2009...

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