Lecture I: The Time-dependent Schrodinger Equation A. Mathematical and Physical Introduction B. Four methods of solution 1. Separation of variables

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

),(),(

txHt

txi

)(2 2

22

xVxm

VTH

1)()(),(),(*

ttdxtxtx

),(),(

txAzt

txz

11 ,, CzRtRx n 11: CRRz n

*AA

1)(),( tztz

),( tx

),(),(* txtx

Physics Perspective: Time-dependent Schrodinger eq.

Integral representation for

Proof of norm conservation

Math Perspective: Complex wave equation

),(),(

txHt

txi

),(),(

txAzt

txz

Physics Perspective: Time-dependent Schrodinger eq.

)0,(),( / xetx iHt

1)0()0(00)()( // iHtiHt eett

)0()( zetz At

1,,, 0000 zzzezezz AtAttt

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

iEtet

Ex

xxVx

m

t

ti

xxVxm

tt

txi

txtx

)0()(

)(

)()(2

)(

)(

)()(2

)()(

)(

)()(),(

2

22

2

22

)()( xExH )0( xas

,nEE

)()( xExH nnn

Solutions of the time-independent Schrodinger equation

particlein a box(discrete)

harmonicoscillator(discrete)

Morse oscillator(discrete +continuous)

IV

Eckart barrier(degenerate continous )

Reconstituting the wavefunction (x,t)

1ΨΨΨ

isTDSEtheofsolutiongeneraltheTherefore,

.ΨΨissosolutionsareΨandΨiflinear,isTDSEtheSince

TDSEtheofsolutionparticularais0Ψ

0

Ψ:ansatzthetoReturning

2

2121

(t)(t)a,(x)eψa(x,t)

)e((x)χψ(x,t)

)eχ((t)χ

(x)ψE(x)Hψ

ψ(x)χ(t)(x,t)

nn

n

tiEnn

tiEnn

tiEn

nnn

n

n

n

Example: Particle in half a box

2

222

8

22 sin

mLn

n

Lxn

Ln

E

/)(),(

)()0,(

tiEn

nn

nn

n

nexatx

xax

Solution of the time-dependent Schrodinger equationMethod 2: Parametrized functional form

For

the ansatz:

leads to the diff eqs for the parameters:

tt

t

tt

tt

tt

mxm

m

p

xmp

m

px

mi

m

i

222

2

2

22

2

1

2

2

2

//)()(exp),( 2ttttt ixxipxxtx

222

22

2

1)(),(

2xmxVxV

xmH

Solution of the time-dependent Schrodinger equationMethod 2: Parametrized functional form

For

the ansatz:

leads to the diff eqs for the parameters:

tt

t

tt

tt

tt

mxm

m

p

xmp

m

px

mi

m

i

222

2

2

22

2

1

2

2

2

//)()(exp),( 2ttttt ixxipxxtx

222

22

2

1)(),(

2xmxVxV

xmH

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

223

3

2

1ˆ

ˆ

ˆˆ

/ˆˆ

ttt

t

t

t

tt

tt

xxx

xV

x

xV

x

xVp

mpx

On intermediate time scales for anharmonic potentials Ehrenfest’stheorem 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

223

3

2

1

/

ttt

t

t

t

tt

tt

xxx

xV

x

xV

x

Vp

mpx

Wigner phase space representation

Wigner phase space representation Harmonic oscillator

Coherent stateSqueezed state Anti-squeezed state

Wigner phase space representation

Particle in half a box

Solution of the time-dependent Schrodinger equationMethod 3: Method of characteristics

Ansatz:

.22

,02

11

),()(),(2

),(

R,,),(exp),(),(

22

2

QA

A

mV

m

SS

ASm

SAm

A

txxVtxm

txi

SAtxSi

txAtx

xxxt

xxxxt

xxt

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”m

Sv x

xv

tdt

d

0 2

2

xxx

t QVdt

dvmQV

m

SS

x

02

2

xxx

t Vdt

dvmV

x

vv

t

vmV

m

SS

x

Classical HJ equation

Classical trajectories

Quantum HJ equation

Quantum trajectories

St Sx

2

2mV

2

2m

Axx

AQ

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.

(x,t)expi

S(x,t)

St Sx

2

2mV i

2mSxx

Complex quantumHamilton-Jacobi equation !

Complex S !

Complex quantum potential

Reconciling Bohm and Ehrenfest

• For Gaussian wavepackets in potentials up to quadratic, the quantum force vanishes!

(x,t)expi

S(x,t)

St Sx

2

2mV i

2mSxx

Complex quantumHamilton-Jacobi equation !

Complex S !

Complex quantum potential

xxx

t Sm

iV

m

SS

x 22

2

dv

dt

Vx

m

i2m

vxx

m

Sv x

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

dxpxxpppdxexx

ipp

dxxxppdxexp

ipx

ipx

ˆˆ)()(~

)()(~

/

/

Digression on the momentum representation and Dirac notation


dxpxxpppdxexx

ipp

dxxxppdxexp

ipx

ipx

ˆˆ)()(~

)()(~

/

/

Digression on the momentum representation and Dirac notation

1,1

dpppdxxx


)0,(

)0(

)0()0()(

:notationDirac

)0,()0,()0,(),(

)(2

:ionapproximatoperatorSplit

/)(/

21

//

///

////)((/

2

22

2

xeFTeFT

xdpdpdexxppeppx

eexextx

xeexexetx

VTxVxm

H

txiVt

m

pi

iVtiTt

iVtiTtiHt

iVtiTtVTiiHt

Increase accuracy by subdividing time interval:

tVtTN

VtTttVtTN

xeeeeee

xeeexetxtiVtiTtiVtiTtiVtiT

tiHtiHtiHiHt

,,.ofinstead,asgoeserror

)0,(

)0,()0,(),(

2

//////

////

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

)actionclassical""theis(2

,

0pathsall

/),,(2/1

1/

2

//

ttxxiSiHt

iHtiHt

LdtSeti

mxex

xdxxexextx

evolution operator or propagator

ti

ti+1

0

t

x’ x


)actionclassical""theis(2

,

0pathsall

/),,(2/1

/

//

ttxxiSiHt

iHtiHt

LdtSeti

mxex

xdxxexextx

evolution operator or propagator

ti

ti+1

0

t

x’ x


)actionclassicaltheis(2

2

)Lagrangianclassicaltheis(22

,,0limittheIn

22

1

0pathsall

/),,(2/1

1/

2

/2/1

path1/

2

/2/1/)(

22/1

22

212

//

2

2/1//

/2/

1//

21/

2

2

2

212

1

2

2

ttxxiSiHt

tLiiHt

tiLtxV

xmi

tiVt

t

xxim

tiVipxt

m

pi

ipx

tiVtiTtiH

LdtSeti

mxex

eti

mxex

VTLeti

me

ti

m

xt

xxΔt

eeti

mdpeeee

xdpdpdxexxppeppxxex

i i

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

nnn

iEt

n

tiEn

iEt

n

tiEnmn

tiEn

nmm

tiEn

nn

iEtn

nn

EEcdteecdtet

ececcdxtxxt

exctx

dtetEEcE

n

nn

n

2//2/

/2/

,

**

/

/2

2

10

2

1

),()0,(0

)(),(

:Derivation

02

1)(

definition :t wavepackea of Spectrum

A.Wavepacket correlation functions for bound potentialsParticle in half a box

A. Wavepacket correlation functions for bound potentialsHarmonic oscillator

A. Wavepacket correlation functions for bound potentials

1

1

1

33

1

22

2

A. Wavepacket correlation functions for unbound potentialsEckart 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

dtetxx tiEn

n

/),()(

eigenfunction wavepacket superposition


)()(2)(),(

)(),(

:Derivation

),()(

///

/

/

xEEcdteexcdtetx

exctx

dtetxx

nnntiEtiE

mm

mtiE

tiEn

nn

tiEn

nnn

n

n



dtetxx tiEn

n

/),()(


n=1E=1.5

2<n<3E=3.0

n=7E=7.5


dtetxx tiEn

n

/),()(


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?

dissociationisomerization

ring opening

CH

OH

C CH

HH

HC

H

C

C CH

O

H

Wavepacket Dancing:Chemical Selectivity by Shaping Light Pulses

1. Review of Tannor-Rice scheme2. Calculus of Variations Approach3. Iterative Approach and Learning Algorithms

(Tannor, Kosloff and Rice, 1985, 1986)

Tannor, Kosloff and Rice (1986)

Optimal Pulse Shapes

)()(lim TTJT

P

J is a functional of : calculus of variations

Formal Mathematical ApproachA. 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

)(

}{

)(

)2()2(

)2()2(

)()(lim][

)],([)(

nn

n

T

dk

TTJ

tt

P

P

EtdtT

2

0)(

)(

)(

)(

)(

)(

)(* t

t

Ht

tH

t

t

ti

b

a

b

a

b

a

Modified Objective Functional

2

00)()()(Re2

)()(lim][

TT

T

tdtti

H

ttdt

TTJ

P

Modified Objective Functional

2

00)()()(Re2

)()(lim][

TT

T

tdtti

H

ttdt

TTJ

P

0J

0)(

t

J

0)(

T

J

(i) (ii) (iii)

abab

it

)(

),(),( TxTx

Ht

i

P

equations of motion for

equations of motion for optimal field

equations of motion for

)()0,( 0 xx

Ht

i

)(T)0(

)0( )(TIterate!

Tannor, Kosloff, Rice (1985-89)Rabitz et al. (1988)

Optimal Control: Iterative Solution

Documents

Lecture I: The Time-dependent Schrodinger Equation A. Mathematical and Physical Introduction B. Four methods of solution 1. Separation of variables