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Lect27_TwoIntPart - Quiz#9 A rigid rotor consists of two...

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Lecture 27: Two Interacting Particles PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Quiz #9 A rigid rotor consists of two masses, m 1 =m 2 =M , attached to the ends of a massless rod of length 2 L . The center of the rod is attached to an axel, so the rod is free to rotate about its center- point. The motion is confined to the x-y plane: 1. What is the classical energy of the rotator in terms of the angular momentum about the center (let the z-axis point out of the page)? 2. What are the quantum energy levels? What is the degeneracy of each level x y
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Review: Motion in a Central Potential A ‘central potential’ is any spherically symmetric potential: Forces are only along radius The Hamiltonian is: Starting from the energy eigenvalue equation: And the definitions: We arrive at the radial wave eq.: The full wavefunction is then: ) ( ) ( r V r V = r ) ( 2 2 2 R V mR L T H r + + = m n m r E m n H m r n , , , , , , , , l l l l = m n m r r n , , , , ) ( , l l l = ! ) ( ) ( 2 ) 1 ( 2 ) ( , 2 2 2 2 2 , , r R r V mr r m r R E n n n l l l l l h h ! ! " # + + + $ $ % & ( = ) ( 1 ) ( , , r R r r n n l l = ! ) , ( ) ( , , , , , 1 ! " ! " m n Y r R r m n r l l l # = Two interacting particles Consider a system of two particles with no external fields By symmetry, the interaction energy can only depend on the separation distance:
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