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Unformatted text preview: Quantum Mechanics: Vibration and Rotation of Molecules 5th April 2010 I. Q. M. Orbital Angular Momentum: Eigenvalues and Eigen- functions of Rigid Rotor Having determined the orbital angular momentum operators for q.m. rota- tion, we will now consider rotation in 3-dimensions (i.e, in a plane). The systems is a rigid two-body rotor with fixed distance between the two masses at each end of the rotor. Using the commutativity of the total angular mo- mentum and one of its components, we will arrive at the eigenfunctions and eigenvalues of the angular momentum operators. Consider Figure 1. The momentum vector is along the z-direction, so let’s determine the eigenvalues and eigenfunctions for the operator corresponding to the z-component of orbital angular momentum. The z-component angular momentum operator, ˆ l z , in Cartesian coordinates is: l z = x p y- y p x . ( classical ) ˆ l z =- i ¯ h x ∂ ∂y- y ∂ ∂x The equivalent expressions for the x- and y- components are: ˆ l x =-...
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