lecture i: the time-dependent schrodinger equation
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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
Solution of the time-dependent Schrodinger equationMethod 4: Numerical methods
dxpxxpppdxexx
ipp
dxxxppdxexp
ipx
ipx
ˆˆ)()(~
)()(~
/
/
Digression on the momentum representation and Dirac notation
1,1
dpppdxxx
Solution of the time-dependent Schrodinger equationMethod 4: Numerical methods
)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
From the the split operator to classical mechanics: Feynman path integration
)actionclassical""theis(2
,
0pathsall
/),,(2/1
/
//
ttxxiSiHt
iHtiHt
LdtSeti
mxex
xdxxexextx
evolution operator or propagator
ti
ti+1
0
t
x’ x
From the the split operator to classical mechanics: Feynman path integration
)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
B. Eigenfunctions as superpositions of wavepackets
)()(2)(),(
)(),(
:Derivation
),()(
///
/
/
xEEcdteexcdtetx
exctx
dtetxx
nnntiEtiE
mm
mtiE
tiEn
nn
tiEn
nnn
n
n
eigenfunction wavepacket superposition
B. Eigenfunctions as superpositions of wavepackets
dtetxx tiEn
n
/),()(
eigenfunction wavepacket superposition
n=1E=1.5
2<n<3E=3.0
n=7E=7.5
B. Eigenfunctions as superpositions of wavepackets
dtetxx tiEn
n
/),()(
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?
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
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