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lecture13_umn - Lecture 13 The Rigid Rotator Microwave...

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Lecture 13 October 9, 2009 The Rigid Rotator Microwave Spectroscopy The Diffuse Interstellar Bands McQuarrie and Simon Chapter 5 pp 173-179 and also Chapter 6
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A Rotating System Two masses connected by a spring: the vibrational Schrödinger equation. Solutions quantum mechanical harmonic oscillator wave functions. Replace the spring with a solid rod (no vibration) and permit the system to rotate about an axis perpendicular to the rod , it will rotate about its center of mass. The kinetic energy for a rotating system is (13-1) where l is the angular momentum and I is the moment of inertia . T = l 2 2 I
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The Moment of Inertia The moment of inertia for a system with multiple particles (13-2) N total particles each having a distinct mass m and distance from the center of mass r . When there are only two particles one can show: (13-3) μ is the reduced mass : (13-4) R=length of the rigid rod connecting them. I = m i r i 2 i = 1 N " I = μ R 2 μ = m 1 m 2 m 1 + m 2 R
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Rigid Rotator in Free Space There is no potential energy affecting the system The Hamiltonian is simply the kinetic energy operator Time-independent, rigid rotator Schrödinger equation for a diatomic molecule (13-5) L 2 : the total angular momentum squared operator (13-6) H " = ( T + V ) " = T " = L 2 2 I " = E " L 2 2 I Y l , m l = l l + 1 ( ) h 2 2 I Y l , m l = E l Y l , m l
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The Rigid Rotator for Molecular Rotation Allowed energies: Quantized by the total angular momentum quantum number l Depend inversely on the moment of inertia Independent of m l m l can take on 2 l + 1 different values : each energy level is 2 l + 1 degenerate . We replace the notation l with J for the total angular momentum quantum number. Define a rotational constant (13-7) L 2 2 I Y l , m l = l l + 1 ( ) h 2 2 I Y l , m l = E l Y l , m l (13-6) B = h 2 2 I
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Energy Levels for the Rigid Rotator Allowed energy levels (13-8) Separation between allowed energy levels depends on B . B depends on atomic masses and R. If identity of the molecule is known (CO), masses known; the only unknown is the bond distance. If we can measure the separation between rotational energy levels (and know which levels are which), we can determine the bond length.
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