Lecture I: The Time-dependent Schrodinger Equation  A. Mathematical and Physical Introduction  B. Four methods of solution 1. Separation of variables 2. Parametrized functional form 3. Method of characteristics 4. Numerical methods

H is a Hermitian operator is a complex wavefunction Normalization has physical interpretation as a probability density A is an anti-Hermitian operator on a complex Hilbert space Inner product on a complex Hilbert space Math Perspective: Complex wave equation Physics Perspective: Time-dependent Schrodinger eq.

Integral representation for  Proof of norm conservation Math Perspective: Complex wave equation Physics Perspective: Time-dependent Schrodinger eq.

Solution of the time-dependent Schrodinger equation Method 1: Separation of variables  Ansatz:  Time-independent Schrodinger eq has solutions that satisfy boundary conditions in general only for particular values of

Solutions of the time-independent Schrodinger equation particle in a box (discrete) harmonic oscillator (discrete) Morse oscillator (discrete +continuous) IV Eckart barrier (degenerate continous )

Reconstituting the wavefunction  (x,t)

Example: Particle in half a box

Solution of the time-dependent Schrodinger equation Method 2: Parametrized functional form  For the ansatz: leads to the diff eqs for the parameters:

Solution of the time-dependent Schrodinger equation Method 2: Parametrized functional form  For the ansatz: leads to the diff eqs for the parameters: Hamilton’s equations (classical mechanics) Classical Lagrangian (Ricatti equattion)

Squeezed state Coherent state Anti-squeezed state

Ehrenfest’s theorem and wavepacket revivals  Ehrenfest’s theorem  Wavepacket revivals On intermediate time scales for anharmonic potentials Ehrenfest’s theorem quite generally breaks down. However, on still longer time scales there is, in many cases of interest, an almost complete revival of the wavepacket and a second Ehrenfest epoch. In between these full revivals are an infinite number of fractional revivals that collectively have an interesting mathematical structure.

Ehrenfest’s theorem and wavepacket revivals  Ehrenfest’s theorem  Wavepacket revivals

Wigner phase space representation

Harmonic oscillator Coherent state Squeezed stateAnti-squeezed state

Wigner phase space representation Particle in half a box

Solution of the time-dependent Schrodinger equation Method 3: Method of characteristics  Ansatz:  LHS is the classical HJ equation: phase  action RHS is the quantum potential: contains all quantum non-locality continuity equation quantum HJ equation

From the quantum HJ equation to quantum trajectories Quantum force-- nonlocal total derivative=“go with the flow” Classical HJ equation Classical trajectories Quantum HJ equation Quantum trajectories

Reconciling Bohm and Ehrenfest The LHS is the classical Hamilton-Jacobi equation for complex S, therefore complex x and p (complex trajectories). The RHS is the quantum potential which is now complex. Complex quantum Hamilton-Jacobi equation ! Complex S ! Complex quantum potential

Reconciling Bohm and Ehrenfest For Gaussian wavepackets in potentials up to quadratic, the quantum force vanishes! Complex quantum Hamilton-Jacobi equation ! Complex S ! Complex quantum potential

Solution of the time-dependent Schrodinger equation Method 4: Numerical methods Digression on the momentum representation and Dirac notation

Solution of the time-dependent Schrodinger equation Method 4: Numerical methods Digression on the momentum representation and Dirac notation

Solution of the time-dependent Schrodinger equation Method 4: Numerical methods

Increase accuracy by subdividing time interval:

From the the split operator to classical mechanics: Feynman path integration evolution operator or propagator titi t i+1 0 t x’x

From the the split operator to classical mechanics: Feynman path integration evolution operator or propagator titi t i+1 0 t x’x

From the the split operator to classical mechanics: Feynman path integration

Lecture II: Concepts for Chemical Simulations  A. Wavepacket time-correlation functions 1. Bound potentials  Spectroscopy 2. Unbound potentials  Chemical reactions  B. Eigenstates as superpositions of wavepackets  C. Manipulating wavepacket motion 1. Franck-Condon principle 2. Control of photochemical reactions

A.Wavepacket correlation functions for bound potentials

Particle in half a box

A. Wavepacket correlation functions for bound potentials Harmonic oscillator

A. Wavepacket correlation functions for bound potentials   

A. Wavepacket correlation functions for unbound potentials Eckart barrier Correlation function and spectrum of incident wavepacket

Correlation function and spectrum of reflected and transmitted wavepackets

Normalizing the spectrum to obtain reflection and transmission coefficients

B. Eigenfunctions as superpositions of wavepackets eigenfunction wavepacketsuperposition

B. Eigenfunctions as superpositions of wavepackets eigenfunction wavepacket superposition

B. Eigenfunctions as superpositions of wavepackets eigenfunction wavepacket superposition n=1 E=1.5 2<n<3 E=3.0 n=7 E=7.5

B. Eigenfunctions as superpositions of wavepackets eigenfunction wavepacket superposition

C. Manipulating Wavepacket Motion  Franck-Condon principle

C. Manipulating Wavepacket Motion  Franck-Condon principle

C. Manipulating Wavepacket Motion  Franck-Condon principle—a second time

C. Manipulating Wavepacket Motion  1. Franck-Condon principle photodissociation

C. Manipulating Wavepacket Motion  Control of photochemical reactions Laser selective chemistry: Is it possible? dissociation isomerization ring opening C H O H CC H H H H C H C CC H O H

Wavepacket Dancing: Chemical Selectivity by Shaping Light Pulses 1. Review of Tannor-Rice scheme 2. Calculus of Variations Approach 3. Iterative Approach and Learning Algorithms (Tannor, Kosloff and Rice, 1985, 1986)

Tannor, Kosloff and Rice (1986)

Optimal Pulse Shapes

J is a functional of  : calculus of variations

Formal Mathematical Approach A. Calculus of Variations (technique for finding the “best shape” (Tannor and Rice, 1985) 1. Objective functional P  is a projection operator for chemical channel  2. Constraint (or penalty) B. Optimal Control (Peirce, Dahleh and Rabitz, 1988) (Kosloff,Rice,Gaspard,Tersigni and Tannor (1989) 3. Equations of motion are added to “deconstrain” the variables

Modified Objective Functional

(i)(ii)(iii)

equations of motion for  equations of motion for optimal field equations of motion for  Iterate! Tannor, Kosloff, Rice ( ) Rabitz et al. (1988) Optimal Control: Iterative Solution