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The. Landau-Teller. model. revisited. Tim Wendler and Manuel Berrondo BYU Physics. I am calculating the dynamics of a collinear atom/diatomic molecule inelastic collision Jacobi coordinates Quantum-Classical coupling Lie algebraic solution 6 coupled equations Canonical ensemble.

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  1. The Landau-Teller model revisited Tim Wendler and Manuel Berrondo BYU Physics

  2. I am calculating the dynamics of a collinear atom/diatomic molecule inelastic collision • Jacobi coordinates • Quantum-Classical coupling • Lie algebraic solution • 6 coupled equations • Canonical ensemble

  3. Collinear Configuration Diatomic molecule Atom • Harmonic oscillator potential for BC • Repulsive interaction for AB • No AC interaction • Energy is in units of • Valid for

  4. Jacobi Coordinates 1 2

  5. Potential Energy Surface

  6. Equations of motion Expansion Classical for Translation Quantum for vibration

  7. Quantum Hamiltonian “dipole” term Expanded and Rearranged Quantum for vibration

  8. Time Evolution Operator Constant ket Quantum equation of motion for vibration

  9. Lie Algebraic Approach Time dependence goes into “c” numbers

  10. Nuances Very general and useful equation End up with terms like Utilize Berrondo anti-symmetric product

  11. 6 coupled equations Not operator equations!

  12. Transition Rates Initial conditions

  13. Phase Space Calculations Initial conditions

  14. Intuitive plot of collision New Trajectory

  15. Classical Comparison Quantum phase gained energy Classical phase lost energy • Same until initial speed passes the max oscillator speed • Certain phase relations result in opposite effects

  16. Mixed States A superposition is in both states A mixture is in perhaps one or perhaps the other No interference A density matrix can represent a statistical mixture of pure states.

  17. Quantum Liouville Equation Initial state in equilibrium at temperature T Just after the collision

  18. Out of equilibrium, for the moment A canonical ensembleof oscillators Non-equilibrium Initial State of canonical ensemble at Temperature T Just after collision, thermal equilibrium is lost

  19. Summary • No wave functions • A simple equation standard • Phase Space • Transition Rates • Canonical Ensemble • Infinite order transitions Specific system input General boson algebra coefficients already calculated!

  20. Future – Reactive Collisions • SN2 Reactions • Nuclear Reactions • Oxidation of methyl esters

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