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

3. Quantum-Classical coupling Lie algebraic solution 6 coupled equations
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

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

5. Jacobi Coordinates 1 2

6. Potential Energy Surface

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

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

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

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

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

12. Not operator equations!
6 coupled equations
Not operator equations!

13. Transition Rates
Initial conditions

14. Phase Space Calculations
Initial conditions

15. Intuitive plot of collision
New Trajectory

16. 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

17. A density matrix can represent a statistical mixture of pure states.
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.

18. Quantum Liouville Equation
Initial state in equilibrium at temperature T
Just after the collision
In principle, isolated quantum systems are very non-ergodic, and one must couple them to the outside world to induce transitions between the many-body Eigenstates needed for equilibrium.

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

20. General boson algebra coefficients already calculated!
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!

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