04082010supplemental - Quantum Mechanics Vibration and...

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Unformatted text preview: Quantum Mechanics: Vibration and Rotation of Molecules 5th April 2010 I. Classical Orbital Angular Momentum and Extension to Q. M. Operators Before considering the eigenvalue equations describing the energetics and eigenfunctions for describing orbital angular momentum in a q.m. sense, we will first consider the operators that are pertinent to measurements of angular momentum on such systems (remembering that operators are asso- ciated with measureables/observables in a q.m. sense). We also note that we will consider angular momentum in general, though the formulation in the following is based on classical analogues, which are based on analogy to orbital motion). We have considered, so far, the translational (particle-in-a- box) and vibrational (harmonic-oscillator) models to describe translational and vibrational dynamical modes as we understand in classical mechanics. With a discussion of angular momentum, we focus on the rotational model of such dynamics to arrive at a complete set of eigenfunctions that we can apply towards describing actual systems, such as atoms and molecules. Consider a particle described by the Cartesian coordinates ( x, y, z ) ≡ r and their conjugate momenta ( p x , p y , p z ) ≡ p The classical definition of the orbital angular momentum of such a par- ticle about the origin is L = r × p giving, L x = y p z- z p y , (1) L y = z p x-...
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This note was uploaded on 02/02/2012 for the course CHEM 444 taught by Professor Dybowski,c during the Fall '08 term at University of Delaware.

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04082010supplemental - Quantum Mechanics Vibration and...

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