Lecture I: The Time-dependent Schrodinger Equation

1. 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

2. 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 Physics Perspective: Time-dependent Schrodinger eq. Math Perspective: Complex wave equation

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

4. Solution of the time-dependent Schrodinger equationMethod 1: Separation of variables • Ansatz: • Time-independent Schrodinger eq has solutions that satisfy boundary conditions in general only for particular values of

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

6. Reconstituting the wavefunction Y(x,t)

7. Example: Particle in half a box

9. Solution of the time-dependent Schrodinger equationMethod 2: Parametrized functional form • For the ansatz: leads to the diff eqs for the parameters:

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

12. Squeezed state Coherent state Anti-squeezed state

13. 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.

14. Ehrenfest’s theorem and wavepacket revivals • Ehrenfest’s theorem • Wavepacket revivals

15. Wigner phase space representation

16. Wigner phase space representation Harmonic oscillator Coherent state Squeezed state Anti-squeezed state

17. Wigner phase space representation Particle in half a box

18. continuity equation quantum HJ equation Solution of the time-dependent Schrodinger equationMethod 3: Method of characteristics • Ansatz: LHS is the classical HJ equation: phase  actionRHSis thequantum potential: contains all quantum non-locality

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

20. Reconciling Bohm and Ehrenfest Complex S ! Complex quantum Hamilton-Jacobi equation ! Complex quantum potential • 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.

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

22. Solution of the time-dependent Schrodinger equationMethod 4: Numerical methods Digression on the momentum representation and Dirac notation

23. Solution of the time-dependent Schrodinger equationMethod 4: Numerical methods Digression on the momentum representation and Dirac notation

24. Solution of the time-dependent Schrodinger equationMethod 4: Numerical methods

25. Increase accuracy by subdividing time interval:

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

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

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

30. 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

31. A.Wavepacket correlation functions for bound potentials

32. A.Wavepacket correlation functions for bound potentials Particle in half a box

33. A. Wavepacket correlation functions for bound potentials Harmonic oscillator

34. t2 t1 t3 A. Wavepacket correlation functions for bound potentials

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

36. Correlation function and spectrum of reflected and transmitted wavepackets

37. Normalizing the spectrum to obtain reflection and transmission coefficients

38. B. Eigenfunctions as superpositions of wavepackets eigenfunction wavepacket superposition

39. B. Eigenfunctions as superpositions of wavepackets eigenfunction wavepacket superposition

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

41. B. Eigenfunctions as superpositions of wavepackets superposition eigenfunction wavepacket

42. C. Manipulating Wavepacket Motion • Franck-Condon principle

43. C. Manipulating Wavepacket Motion • Franck-Condon principle

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

45. C. Manipulating Wavepacket Motion • 1. Franck-Condon principle photodissociation

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

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

49. Tannor, Kosloff and Rice (1986)