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3502_HWkey4 - Chem 3502/5502 Physical Chemistry II(Quantum...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Answers to Homework Set 4 From lecture 13: Consider a 2-dimensional so-called planar rigid rotator—a quantum mechanical compact disc, if you will. In this system, rotation is confined to a plane, so all of the angular momentum is along a single axis. The Schrödinger equation for this system is thus L z 2 2 I " = E " If the moment of inertia I is taken to be 1/2, what are the eigenfunctions and eigenvalues for this system (use spherical polar coordinates)? Looking at the Schrödinger equation for the free particle may be helpful, but this case is quantized, while that for the free particle is not—why is there a difference? What are the lowest 3 possible energies? What degeneracies are associated with these energies? We have the restricted Schrödinger equation L z 2 2 I " = E " or, noting that I = 1/2 and using eq. 12-8 for L z " h 2 d 2 d # 2 $ # ( ) = E $ # ( ) where φ is the variable describing rotation of the disk from 0 to 2 π . This equation can be rearranged to d 2 d " 2 + E h 2 # $ % & ( ) " ( ) = 0 which looks almost identical to the free particle wave function but with different constants, and has the solutions (eigenfunctions) " # ( ) = Ce i E # / h
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HW4-2 For the free particle, all values of x
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