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

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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 Schrdinger Equation: Spherical Polar Coordinates Rotational Spectroscopy of Linear Molecules Not Last Topic Chapter 4: Slide 3 Outline (Contd.) 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 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( B A B A = Magnitude: Parallel Vectors: ) cos( B A B A = B A = Chapter 4: Slide 5 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( 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 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 Schrdinger Equation: Spherical Polar Coordinates Rotational Spectroscopy of Linear Molecules Chapter 4: Slide 8 Rotational Motion in Classical Physics p r L = ) sin( rp L = Magnitude: Angular Momentum (L) m r p Circular Motion: rp rp L = = ) 90 sin( or: ( 29 ) ( 2 2 mr r v mr rmv rp L = = = = I L = where r v mr I = = 2 Energy 2 2 2 2 mv m p E = = 2 2 ) ( 2 2 2 mr r m = = 2 2 I = or: I L I I E 2 2 ) ( 2 2 = = 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 = Angular velocity v Momentum I L = Angular momentum p=mv Energy I L E 2 2 = Energy m p E 2 2 = or 2 2 I E = Energy 2 2 mv E = Chapter 4:...
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5210-chap4 - Chapter 4: Slide 1 Chapter 4 Rigid-Rotor...

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