lecture_23_Ang_mom1

lecture_23_Ang_mom1 - Chem 120A QM treatment of Angular...

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Unformatted text preview: Chem 120A QM treatment of Angular Momentum 03/14/07 Spring 2007 Lecture 23 READING: Ratner and Schatz: Chapter 4 Atkins and Friedman: Chapter 3.1-3.2 Rotating diatomic molecule Angular momentum arises in the study of rotating diatomic molecules. Suppose diatomic molecule consists of two atoms with masses m 1 and m 2 and the separation between these two atoms is r . Suppose this molecule rotates about the center of mass R . Let the distance from m 1 to R be r 1 and the distance from m 2 to R be r 2 . Then linear velocity of m 1 is v 1 = r 1 ϖ and linear velocity of m 2 is v 2 = r 2 ϖ , where ϖ is the angular frequency (rad/s) for the rotation. Classical approach We can calculate the kinetic energy for rotation K . E . = 1 2 m 1 v 2 1 1 2 m 2 v 2 2 = 1 2 ( m 1 r 2 1 + m 2 r 2 2 ) ϖ 2 = 1 2 I ϖ 2 , (1) where I = m 1 r 2 1 + m 2 r 2 2 is moment of inertia for the diatomic molecule. Now the center of mass R is given by m 1 r 1 = m 2 r 2 and r = r 1 + r 2 . So m 1 r 2 1 + m 2 r 2 2 = m 2 r 2 ( r 1 + r 2 ) = m 2 r 2 r 1 + r 2 r 2 = bracketleftbigg r 1 m 2 r 2 + 1 m 2 bracketrightbigg- 1 r 2 = bracketleftbigg 1 m 1 + 1 m 2 bracketrightbigg...
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lecture_23_Ang_mom1 - Chem 120A QM treatment of Angular...

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