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

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

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

Jacobi Coordinates 1 2

Potential Energy Surface

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

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

Time Evolution Operator Constant ket Quantum equation of motion for vibration

Lie Algebraic Approach Time dependence goes into “c” numbers

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

Not operator equations! 6 coupled equations Not operator equations!

Transition Rates Initial conditions

Phase Space Calculations Initial conditions

Intuitive plot of collision New Trajectory

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

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.

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.

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

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!

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