5210-chap4

# 5210-chap4 - Chapter 4 Slide 1 Chapter 4 Rigid-Rotor Models...

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Unformatted text preview: Chapter 4: Slide 1 Chapter 4 Rigid-Rotor Models and Angular Momentum Eigenstates Chapter 4: Slide 2 Outline • Math Preliminary: Products of Vectors • Rotational Motion in Classical Physics • The 3D Quantum Mechanical Rigid Rotor • Angular Momentum in Quantum Mechanics • Angular Momentum and the Rigid Rotor • The 2D Quantum Mechanical Rigid Rotor • The 3D Schrödinger Equation: Spherical Polar Coordinates • Rotational Spectroscopy of Linear Molecules Not Last Topic Chapter 4: Slide 3 Outline (Cont’d.) • Application of QM to Molecular Structure: Pyridine • Statistical Thermodynamics: Rotational contributions to the thermodynamic properties of gases Chapter 4: Slide 4 Mathematical Preliminary: Products of Vectors θ B A k A j A i A A z y x + + = k B j B i B B z y x + + = Scalar Product (aka Dot Product) z z y y x x B A B A B A B A + + = • Note that the product is a scalar quantity (i.e. a number) ) cos( θ B A B A = • Magnitude: Parallel Vectors: ) cos( B A B A = • B A = Chapter 4: Slide 5 θ B A k A j A i A A z y x + + = k B j B i B B z y x + + = Cross Product The cross product of two vectors is also a vector. Its direction is perpendicular to both A and B and is given by the “right-hand rule”. ) sin( θ B A B A = × Magnitude: Parallel Vectors: ) sin( B A B A = × = B A × B A = ) 90 sin( B A B A = × Perpendicular Vectors: Chapter 4: Slide 6 θ B A k A j A i A A z y x + + = k B j B i B B z y x + + = B A × z y x z y x B B B A A A k j i B A = × y x y x z x z x z y z y B B A A k B B A A j B B A A i B A +- = × Expansion by Cofactors k B A B A j B A B A i B A B A B A x y y x z x x z y z z y ) ( ) ( ) (- +- +- = × Chapter 4: Slide 7 Outline • Math Preliminary: Products of Vectors • Rotational Motion in Classical Physics • The 3D Quantum Mechanical Rigid Rotor • Angular Momentum in Quantum Mechanics • Angular Momentum and the Rigid Rotor • The 2D Quantum Mechanical Rigid Rotor • The 3D Schrödinger Equation: Spherical Polar Coordinates • Rotational Spectroscopy of Linear Molecules Chapter 4: Slide 8 Rotational Motion in Classical Physics p r L × = ) sin( θ rp L = Magnitude: Angular Momentum (L) m r p θ Circular Motion: rp rp L = = ) 90 sin( or: ( 29 ϖ ) ( 2 2 mr r v mr rmv rp L = = = = Iω L = where r v mr I = = ϖ 2 Energy 2 2 2 2 mv m p E = = 2 2 ) ( 2 2 2 ϖ ϖ mr r m = = 2 2 ϖ I = or: I L I I E 2 2 ) ( 2 2 = = ϖ Moment of Inertia Angular Frequency Chapter 4: Slide 9 Comparison of Equations for Linear and Circular Motion Linear Motion Circular Motion Mass 2 mr I = Moment of inertia m Velocity r v = ϖ Angular velocity v Momentum ϖ I L = Angular momentum p=mv Energy I L E 2 2 = Energy m p E 2 2 = or 2 2 ϖ I E = Energy 2 2 mv E = Chapter 4:...
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## This note was uploaded on 04/11/2011 for the course CHEM 5210 taught by Professor Staff during the Spring '08 term at North Texas.

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5210-chap4 - Chapter 4 Slide 1 Chapter 4 Rigid-Rotor Models...

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