{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture13_umn

# lecture13_umn - Lecture 13 The Rigid Rotator Microwave...

This preview shows pages 1–7. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}