Download 2 Quantum dynamics of simple systems

2 Quantum dynamics of simple systems
2.1 Time-dependent Schrödinger equation and time-evolution
operator
In spectroscopy, kinetics and scattering, time-dependent problems need to be solved.
They are described by Time-dependent Schrödinger equation (TDSE):
∂Ψ(x, y, .....xn , yn , zn , t)
= ĤΨ(x, y, .....xn , yn , zn , t).
∂t
The general solution of the TDSE is:
i
Ψ(t) = Û (t, t0 )Ψ(t0 ).
(2.1)
(2.2)
The time-evolution operator is obtained by solving the equation:
i
∂ Û (t, t0 )
= Ĥ Û (t, t0 ).
∂t
(2.3)
The special case of a time-independent Ĥ leads to
Û (t, t0 ) = e
−
iĤ(t−t0 )
,
(2.4)
Q̂n
where eQ̂ = ∞
n=0 n! . can be used for the evaluation.
The stationary solutions are given by:
ĤΨ(r, t) = ÊΨ(r, t) = EΨ(r, t).
Since the time dependence of the stationary solutions is given by e
i
−iEt
∂Ψ(r, t)
= ÊΨ(r, t) = EΨ(r)e .
∂t
(2.5)
−iEt
it follows that
(2.6)
Using the relation
∂ −iEt
−iEt −iEt
e =
e ,
∂t
We obtain the time-independent Schrödinger equation (TISE)
ĤΨk (r, t) = Ek Ψk (r, t)
The eigenfunctions ψk (r, t) = ψk (r)e
−iEk t
6
(2.7)
(2.8)
are also called the ”stationary states”.
2 Quantum dynamics of simple systems
2.1.1 General time-dependent states
Any linear combination of solutions of the TDSE is also a solution
Ψ(r, t) = c1 Ψ1 (r, t) + c2 Ψ2 (r, t),
(2.9)
or in general we can write:
Ψ(r, t) =
ck Ψk (r)e
−iEk t
.
(2.10)
k
this case both
When Ĥ is time independent then the ck are also time independent. In 2
|ck |2 Ek are
the energy distribution pk (Ek ) = |ck | and the average energy E(t) =
time independent. If Ĥ is time dependent, then the ck (t) are time-dependent and both
the energy distribution and the average energy will change with time.
2.1.2 Probability densities
The probability density is defined by:
|Ψk (r, t)| = ψk (r)e
2
−
iEk t
ψk∗ (r)e
iEk t
(2.11)
= |ψk (r)|2
Equation (2.11) shows that the probability density is independent of time and therefore
any solution of the TISE is stationary. Now consider the linear combination
Ψ(r, t) = aΨ1 (r)e
−iE1 t
+ bΨ2 (r)e
−iE2 t
.
(2.12)
The probability density is given by:
|Ψ(r, t)|2 = |a|2 |Ψ1 (r)|2 + |b|2 |Ψ2 (r)|2 + 2Re (a∗ bΨ∗1 Ψ2 ) cos
(E2 − E1 )t
.
(2.13)
The third term in equation (2.13) is an interference term that contains all of the time
dependence of |Ψ(r, t)|2 .
In the most general case (for example in atoms) both discrete and continuous spectra
need to be included:
Ψ(r, t) =
∞
ck Ψk (r)e
−iEk t
∞
+
0
n=1
7
a(E)ΨE (r)e
−iEt
dE.
(2.14)
2 Quantum dynamics of simple systems
2.2 Dynamics of the harmonic oscillator
The stationary harmonic-oscillator problem has already been treated in PC III. The
Hamiltonian is given by
2 d 2
1
+ mω 2 x2 ,
(2.15)
2m dx2 2
Where m is the mass and ω is the angular frequency of the oscillator. We do not give
the explicit coordinate dependences and write ϕn for ϕm and Ψ(t) for Ψ(r, t).
Consider the superposition of the 2 lowest quantum states
Ĥ = −
−iE0 t
−iE1 t
1
Ψ(t) = √ (ϕ0 e + ϕ1 e )
2
−i(E1 −E0 )t
1 −iE0 t
)
= √ e (ϕ0 + ϕ1 e
2
(2.16)
1
|Ψ(t0 )|2 = (ϕ0 + ϕ1 )2
2
(2.17)
1
|Ψ(t1 )|2 = (ϕ0 − ϕ1 )2
2
(2.18)
1
|Ψ(t2 )|2 = (ϕ0 + ϕ1 )2
2
(2.19)
At t0 = 0
At t1 =
At t2 =
h
2(E1 −E0 )
h
(E1 −E0 )
h
The oscillation period is T = ΔE
.
This reslut can be generatized to N +1 states of the harmonic oscillator. The energy
level are given by:
En = E0 + nδ.
(2.20)
A general time-dependent wave function is:
Ψ(r, t) = e
−iE0 t
N
cn ϕn (r)e
−inδt
,
(2.21)
n=0
which has a period τ = hδ = tτ . At every time t = kτ, k = N , all phases are identical
(module 2π). The wave packet is therefore at all these times. The situation is called
a ”revival”. In the special case of equidistant spectra the revival (a recurrence)time is
equal to period.
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2 Quantum dynamics of simple systems
Figure 2.1: Tunneling and ammonia inversion
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2 Quantum dynamics of simple systems
Figure 2.2: Separated potential wells (no tunneling).
2.3 Tunneling and ammonia inversion
We first consider states in the localized left or right potential minimum ϕl , ϕr . The lowest
lying stationary states have energies E1 − E0 = ΔE. The corresponding wavefunctions
are
1
ϕ0 √ (ϕl + ϕr )
2
1
ϕ1 √ (ϕl − ϕr ).
2
(2.22)
We assume that at t0 the wavepacket is localized in the left well.
1
Ψ(t0 ) = √ (ϕ0 + ϕ1 ) = ϕl
(2.23)
2
This is not an eigenstate (stationary solution) but a superposition state or wavepacket
(identical to equation (2.16)).
−i(E1 −E0 )t
1 −iE0 t
).
Ψ(t) = √ e (ϕ0 + ϕ1 e
2
Again, for t1 =
h
2(E1 −E0 )
(2.24)
the wavefunction becomes
1
Ψ(t1 ) = √ (ϕ0 − ϕ1 )
2
(2.25)
h
which gives the oscillation period T = ΔE
. This relation is a general result for 2-state
wavepacets.
−1 and the oscillation period is T = h =41.9
For ammonia in J =0, ΔE
c = 0.796 cm
ΔE
ps. The splitting ΔE depends on J and lies between 0.55 and 1.3 cm−1 . Vibrationallyexcited levels have much larger ΔE (because of the lower barrier to tunneling). The
transition between the two lowest energy levels of ammonia is dipole allowed and has been
used to realize the first MASER (Charles Townes, 1951). NH3 in the lower tunnelling
level is deflected away by an inhomogeneous electric field. This results in a population
inversion of the ammonia molecules. The hollow-resonator cavity is tuned to Ω = 2πν =
2π
T . Molecules emit microwave radiation then leave the cavity.
2.4 Multilevel quantum beats and revivals
2.4.1 Alignment and orientation of molecules
Rigid linear rotor: stationary problem treated in PC III.
Ĥ = cB Jˆ2
10
(2.26)
2 Quantum dynamics of simple systems
Figure 2.3: Potential wells connected by a low barrier (tunneling).
Figure 2.4: Realization of the MASER
The total energy of the molecule in a static electric field is:
E = E(0) + (
d3 E
dE
1 d2 E
1
)i + (
)i j + (
)i j k + ......
di
2 di dj
6 di dj dk
(2.27)
with summation over repeated indices implied.
This expression can be rewritten in terms of the induced dipole moment μi
E = E (0) − μi i
(2.28)
1
μi = μ0,i + αij j + βijk j k + ....
2
(2.29)
where
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2 Quantum dynamics of simple systems
with the following definations
Permanent dipole moment:
Polarizability terms:
Hyperpolarizability terms:
dE
)
di
d2 E
)
αij = −(
di dj
d3 E
βijk = −(
)
di dj dk
μ0,i = −(
1
Ĥ = cB Jˆ2 − μ
· − · α · + ....
2
Total Hamiltonian:
(2.30)
(2.31)
Molecular alignment can be realized in:
1 )Static fields: The interaction of a static field with the permanent dipole moment
μ of the molecule. This techniques is known as ”brute-force orientation” (for polar
molecules μ = 0).
2
)Oscillating fields: (non-resonant)
= ez 0 cos(ωt)
μ0 · Udipole (θ) = −
= −μ0 0 cos θ cos(ωt)
−→ Averages to zero over each optical cycle
(2.32)
The polarizibilty tensor is diagonal in molecular
⎛
αxx 0
αmol = ⎝ 0 αyy
0
0
frame (MF)
⎞
0
0 ⎠
αzz
(2.33)
For linear molecules to which we restrict our considerations here:
αxx = αyy = α⊥
and αzz = α
(2.34)
Therefore,
1
Uind (θ, φ) = − · α · 2
1 2
= − 0 cos2 (ωt)[(α − α⊥ ) cos2 θ + α⊥ ].
2
(2.35)
1
Ūind (θ) = − 0 2 [(α − α⊥ ) cos2 θ + α⊥ ].
4
(2.36)
After averaging over an oscillating field:
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2 Quantum dynamics of simple systems
Figure 2.5: Stimulated Raman transitions in a linear rigid rotor.
The hyperpolarizibilty interaction can be used to orient polar molecules but is ususally
much weaker than the polarizability interaction. We theofore continue with
1
(2.37)
Ĥrot = B Jˆ2 − 2 (Δα cos2 θ + α⊥ ).
4
Short and intense laser pulses create multiple stimulated-Raman transitions that follow
ΔJ = 0, ±2 in closed-shell linear molecules. Starting from J =0 we obtain a broad, fully
coherent superposition state In the case of the rigid rotor, the period of the wavepacket
motion is equal to the revival time.
|Ψ =
cJ |JM e−
iEJ t
(2.38)
J
YJM (θ, φ)
are spherical harmonics, i.e. eigenfunctions of the linear
where θ, φ|JM =
rigid rotor. The energy eigenvalues are given by:
EJ = hcBJ(J + 1).
Thefore, the period of the wave-packet evolution is given by:
1
h
=
.
T =
ΔE
2Bc
(2.39)
Useful measure is the ”degree of axis alignment” cos2 θ where
π
cos2 θ =
0
2π
Ψ∗ (θ, φ) cos2 (θ)Ψ(θ, φ) sin θdθdφ
0
13
(2.40)
2 Quantum dynamics of simple systems
Figure 2.6: Coordinate system defining the spatial orientation of a linear molecule.
In all practical cases, the initial state is a thermal ensemble with ”a rotational temperature” Trot , i.e. a Boltzmann-like population distribution. In this case we need a
density matrix ρ to describe the quantum system. The wave packet evolved from each
initial state |J0 M0 is
J ,M
CJ 0 0 (t)|JM0 (2.41)
|φJ0 ,M0 =
J
Then the density matrix is
ρ(t) =
pJ0 |φJ0 ,M0 (t)φJ0 M0 (t)|,
(2.42)
J0 ,M0
Where pJ0 is the initial population of level Jo ; The coefficients CJJ0 M0 (t) are obtained
by solving the TDSE. The temporal evolution of cos2 θ of N2 molecules is shown in
figure 2.7 assuming an 800 nm, 48 fs laser pulses and a rotational temperature of 30 K.
Anharmonic Oscillator:
Figure (2.8) shows the Morse potential which was treated in PC III and PC V:
V (R) = De (1 − e−a(R−Re ) )2
The eigenvalues are given by:
1
1
h2 ν02
(n + )2
En = hν0 (n + ) −
2
4De
2
1
1
= hν0 [(n + ) − α(n + )2 ],
2
2
(2.43)
(2.44)
where α is the anharmonicity parameter.
We consider an initial wavepacket (r is the internuclear separation):
Ψ(r, t) =
cn χn (r)e−
n
14
iEn t
h
(2.45)
2 Quantum dynamics of simple systems
Figure 2.7: Temporal evolution of cos2 θ that indicates the alignment of N2 molecules
initially at 30 K rotational temperature, excited using an 800 nm, 48 fs pulse
of peak intensity 5×1013 W/cm2 .
Revival time:
The wavepacket is fully reconstructed with original phases at:
2π
ω0 α
Trev =
(2.46)
We introduce the autocorrelation function:
P (t) = Ψ(r, t)|Ψ(r, 0)
−iEn t
=
|cn |2 e (2.47)
n
Which is normalizaed such that P (0) = n |cn |2 = 1. We now consider the anharmonic Morse oscillator with a redefined energy scale:
Gn = hν0 (n − αn2 ).
(2.48)
Assuming a Gaussian population distribution of the vibrational levels, centered at ν0 ,
1
|cn |2 = (πγ)− 2 e
= (πγ)
[−
(n−n0 )2
−ihν0 nt+ihν0 αn2 t]
γ
− 12 (−Gn0 t)θ3 (u|τ )
e
where θ3 (u|τ ) is the Jacobi theta function
1
(2.49)
1
see Whittaker and Watson, A course of modern analysis, 1996, Chapter 21, Cambridge University
Press
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2 Quantum dynamics of simple systems
Figure 2.8: A Morse potential compared to the corresponding harmonic potential and
schematic illustration of the energy eigenvalues.
.
θ3 (u|τ ) =
,
∞
n=−∞
16
e(iπτ n
2 +2inu)
(2.50)
2 Quantum dynamics of simple systems
Properties of θ3 :
1. Bi-periodicity:
a) θ3 (u + π | τ ) = θ(u | τ )
b) θ3 (u | τ + 2) = θ3 (u | τ )
2. Quasi-periodicity in u: θ3 (u+τ ) = e(iπτ k
integer.
2 +2ikn)
1
θ3 (u+πkτ ), with k arbitrary
μ2
3. Jacobi transformation: θ3 (u | τ ) = (−iτ )− 2 e( iπτ ) θ3 ( uτ | − τ1 )
and
u=
hν0
− hν0 αn0 t
2
1
= En 0 t
2
dEn
En 0 =
|n=n0
dn
hν0 αt
i
+
and
τ=
πγ
π
(2.51)
1. Fast variable μ is the phase of the classical motion described by the center
. Tn0 is the classical period.
of inertia of the wavepacket u = Tπt
n
0
2. Slow variable τ describes the dispersion of wave-packet center
2.4.2 General formulation of fractional revivals
We have discussed the Ehrenfest theorem which states that the expectation values x
and p follow classical dynamics for short times. On much longer time scales the wave
packet revives. In systems with quasi-equidistant levels (e.g Rydberg states or levels of
an anharmonic oscillator) revivals occur at early times. We assume the general eigenstate
representation of a wavepacket
−iEt
cn χn (x)e .
(2.52)
Ψ(x, t) =
n
For large quantum numbers n we can use a Taylor expansion
En = En̄ +
1 d2 E
dE
(n − n̄) +
(n − n̄)2 + ....
dn̄
2 dn̄2
17
(2.53)
2 Quantum dynamics of simple systems
And define the classical action J = n̄h. Then
dE
h2 d 2 E
(n − n̄)2 + ....
En = En̄ + h ¯ (n − n̄) +
2 dJ 2
dJ
(2.54)
The classical relation
dE
1
= νcl =
.
dJ
Tcl
Inserting them into Eq (2.54) yields:
(2.55)
En = En̄ + hνcl (n − n̄) +
En̄ + h(
where
h2 d2 νcl
(n − n̄)2 + ....
2 dJ 2
k
k2 2
+
)
Tcl Trev
k = n − n̄
= 0, ±1, ±2, .....
−1
2 d2 E
Trev =
h dJ 2
dνcl 2
2
)
= (νcl
h
dE
dνcl dE
dνcl
d2 E
dνcl
=
=
νcl
=
2
dJ
dJ
dE dJ
dE
where we have used
(2.56)
Keeping only the linear terms yields:
Ψ(x, t) ≈ Ψcl (x, t) ≡
ck χk (x)e
− 2πikt
T
cl
k
= Harmonic Oscillator result
Valid for
t <<
(2.57)
Trev
.
(Δn)2
Keeping both the linear and quadratic terms yields:
Ψ(x, t) ≈
ck χk (x)e
2t
rev
−2πi( Tkt + Tk
cl
)
(2.58)
k
At t =
m
n Trev
(m,n mutually prime integers)
Ψ(x, t) =
ck χk (x)e
−2πi( Tkt +θk )
cl
(2.59)
k
2
where θk = { m
n k } is the fractional part of the argument e.g. {0.7}=0.7, {2.3}=0.3,
etc. From this result one can show that
Ψ(x, t) =
n−1
as Ψcl (x, t +
s=0
18
s
Tcl )
n
(2.60)
2 Quantum dynamics of simple systems
Exercise: Derive equation (2.60).
The wave packet is thus made up of replicas of the t=0 wavepacket shifted by ns of
the classical period. The number of replicas r is equal to the number of non-vanishing
as .
2.5 Correlation functions and spectra
A bound state problem allows expansion in eigenfunctions:
−iEn t
cn Ψn (x)e Ψ(x, t) =
(2.61)
n
The spectrum of the wave packet is:
σ(ω) =
|cn |2 δ(ω − ωn )
(2.62)
n
The spectrum can also be calculated as Fourier transform of the auto-correlation
function (as introduced before)
∞
1
Ψ(0)|Ψ(t)eiωt dt
(2.63)
σ(ω) =
2π −∞
Exercise: Insert (2.61) into (2.63) to obtain (2.62).
The spectrum σ(ω) and auto-correlation function C(t) = Ψ(0)|Ψ(t) are Fourier pairs.
This result is not limited to discrete levels but can be generalized to include continua.
There are three time scales:
1. T1 describes the decay of the auto-correlation function as the wave packet moves
away from its initial location (e.g the Frank-Condon point)
2. T2 describes the recurrence as the wave packet returns to its initial location for the
first time.
3. T3 describes the decay of the recurrence amplitude. Possible reasons include:
a) anharmonicity
b) coupling to other degrees of freedom.
c) irreversible decay due to predissociation, radiationless or radiative decays etc.
There could be T4 for later recurrence in the absence of irreversible decay (revivals)
Consequences:
1. The broader a spectrum is, the faster the decay (T1 ).
2. Periodic spectral structure always leads to periodicity in the auto-correlation function c(t).
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2 Quantum dynamics of simple systems
Figure 2.9: (a) Schematic diagram showing three time scales in the auto-correlation function, (b) corresponding three time scales in the energy spectrum (taken from
Introduction to quantum mechanics, David J. Tannor)
Figure 2.10: Correlation function with a series of three recurring peaks (a), corresponding spectrum of the correlation function (b) showing interference pattern
and ringing from sharp truncation of the last peak in (a)(taken from Introduction to quantum mechanics, David J. Tannor)
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2 Quantum dynamics of simple systems
Figure 2.11: Correlation function given by the product of a slow Gaussian decay and a
series of three Gaussian peaks (a), and corresponding spectrum (b). Note
that the structure of Fig.2.10 remains while the ringing has disappeared.
(taken from Introduction to quantum mechanics, David J. Tannor)
2.6 Solution of TDSE
What we have used so far is called the ”Schrödinger picture”.
A = Ψ(t)|Â|Ψ(t)
e.g:
= e
−iĤt
Ψ(0)|Â|e
−iĤt
(2.64)
Ψ(0).
Properties:
1. Operators are stationary (time independent)
2. State vector Ψ and expectation values evolve in time.
We now turn to
The ”Heisenberg picture”
in which the wavefunction ΨH is defined by:
Since
We find a
ΨH (t) = e
iĤt
ΨS (t) = e
−iĤt
ΨS (t)
ΨS (0)
(2.65)
ΨH (t) = ΨS (0) = ΨH (0)
Operators in the Heisenberg picture ÂH are related to their counterparts in the
Schrödinger picture  by
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2 Quantum dynamics of simple systems
ÂH (t) = e
iĤt
Âe−
iĤt
.
Âe−
iĤt
e
(2.66)
Properties:
1. State vector Ψ is stationary.
2. Operators are time dependent.
A = ΨS (t)|Â|ΨS (t)
= ΨS (t)|e−
iĤt
e
iĤt
iĤt
|ΨS (t)
(2.67)
= ΨH (0)|ÂH (t)|ΨH (0)
For the special case of time-independent Hamiltonians:
i
d
ÂH = [ÂH , Ĥ].
dt
(2.68)
For the special case of position and momentum operators q̂ and p̂
[q̂H , Ĥ] = i
p̂H
m
[p̂H , Ĥ] = −i
dĤ
dq
(2.69)
H
hence
dp̂H
∂ Ĥ
∂ V̂
=−
=−
dt
∂qH
∂qH
∂ Ĥ
dq̂H
p̂H
=
=
dt
∂pH
m
(2.70)
Have discussed the Schrödinger picture and Heisenberge picture, we now turn to:
The interaction picture that is particularly useful when the Hamiltotian can be
written as:
Ĥ = Ĥ0 + V̂ , where the time evolution generated by Ĥ0 is known
ΨI (t) = e
iĤ0 t
ΨS (t)
=e
iĤ0 t
e−
iĤt
(2.71)
ΨS (0)
If time evolution is dominated by Ĥ0 , the forward propagation in time under Ĥ is
nearly canceled by the backward propagation under Ĥ0 , hence ΨI (t) evolves much less
than ΨS (t)(see Fig. 2.12).
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2 Quantum dynamics of simple systems
Figure 2.12: Qualitative diagram showing the relationship between the Schrödinger and
interaction picture wavefunctions propagating on a purely repulsive potential (taken from Introduction to quantum mechanics, David J.Tannor)
The equation of motion for ΨI (t) is:
i
iĤ0 t
iĤ0 t
∂ΨI (t)
= [e V̂ e− ]ΨI
∂t
= ĤI (t)ΨI (t)
(2.72)
This result can be derived as follows:
∂ΨS (t)
= (Ĥ + V̂ )ΨS (t)
∂t
iĤ0 t
iĤ0 t
iĤ0 t ∂Ψ
−iĤ0 t
∂e− ΨI (t)
I
= i
e− ΨI (t) + ie− i
∂t
∂t
i
left-hand side:
(Ĥ0 + V̂ )e−
right-hand side:
Multiplying both with e
iĤ0 t
iĤ0 t
= Ĥ0 e−
iĤ0 t
ΨI (t) + ie−
iĤ0 t
ΨI (t) = Ĥ0 e−
iĤ0 t
ΨI (t) + V̂ e−
iĤ0 t
∂ΨI
∂t
ΨI (t)
from the left yields:
iĤ0 t
−iĤ0 t
∂ΨI (t)
i
= [e V̂ e ]ΨI
∂t
= HI ΨI (t)
(2.73)
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2 Quantum dynamics of simple systems
The set of eigenfunctions
{ψn } of Ĥ fullfill Ĥ0 ψn = En0 ψn .
We expand ΨI (t) = n an (t)ψn and insert this into Eq.(2.73) of motion to find:
iȧm (t) =
Vmn e
i(Em −En )t
an (t)
(2.74)
n
where Vmn = Ψm |V̂ |Ψn .
In the interaction picture {am (t)} are slowly varying.
Defining Hmn (t) = Vmn e
interaction picture
i(Em −En )t
we obtain a matrix representation of the TDSE in the
iȧm (t) =
Hmn (t)an (t).
(2.75)
n
We summarize the following results:
1. Schrödinger picture: Only the wavefunction evolves in time.
2. Heisenberg picture : Only the operators evolve in time.
3. Interaction picture: both wavefunction and operators contain a part of the
time evolution.
The relation between the three pictures is given by the following equations:
A = ΨS (t)|Â|ΨS (t)
= ΨS (0)|e
iĤt
ÂS e
−iĤt
|ΨS (0)
= ΨH |AH (t)|ΨH = ΨS (0)|e
iĤt
e
−iĤ0 t
(2.76)
e
iĤ0 t
ÂS e
−iĤ0 t
e
iĤ0 t
e
−iĤt
|ΨS (0)
= ΨI (t)|AI (t)|ΨI (t)
2.7 Time-dependent perturbation theory
The interaction picture, described in the previous section, is formally exact. In this
section we introduce time-depent perturbation theory. We assume that the Hamiltonian can be written as Ĥ(t) = Ĥ0 + Ĥ1 (t). It is used to develop a series of sucessive
approximations to the evolving wavefunctions. Time-dependent perturbation theory is:
• Useful if dynamics under Ĥ0 are known
• Only recommended if V <<H 0
• Result can be derived using an ordering parameter λ (see textbooks) or from
interaction picture. We use the latter derivation here.
24
2 Quantum dynamics of simple systems
The time propagator associated with Ĥ0 is given by
i
∂ (0)
Û (t, t0 ) = Ĥ0 Û (0) (t, t0 )
∂t
(2.77)
The interaction-picture propagator is
ÛI (t, t0 ) = e
iĤ0 (t−t0 )
e
−iĤ(t−t0 )
(2.78)
= Û (0)† (t, t0 )Û (t, t0 )
The interaction picture propagator satisfies
i
∂
ÛI (t, t0 ) = ĤI (t)ÛI (t, t0 )
∂t
ĤI (t) = e
where
iĤ0 t
Ĥ1 (t)e
−
(2.79)
iĤ0 t
Equation (2.79) can be solved iteratively yielding
ÛI (t, t0 ) = 1 +
∞
(n)
ÛI (t, t0 )
n=1
where
Û
(n)
1
(t, t0 ) =
(i)n
with ordering
t
dτn
t0
τn
dτn−1 .......
t0
τ2
t0
dτ1 ĤI (τn )ĤI (τn−1 )......ĤI (τ1 )ÛI (τ1 , t0 )
t > τn > τn−1 > ..... > τ1 > t0 .
(2.80)
Perturbation series: Replacing ÛI (τ1 , t0 ) with 1, i.e. Ĥ(t) = Ĥ0 leads to
Û (t, t0 ) = Û
(0)
(t, t0 ) +
∞
Û (n) (t, t0 ),
(2.81)
n=1
where
Û
(n)
1
(t, t0 ) =
(i)n
t
to
dτn
t0
(0)
× Ĥ1 (τn−1 ).......Û
Substituting Û (0) (t, t ) = e
ψ
ψ
(1)
(2)
τn
dτn−1 .....
t
t0
1
(t) =
(i)2
t0
dτ1 Û (0) (t, τn )H1 (τn )Û (0) (τn , τn−1 )
(τ2 , τ1 )Ĥ1 (τ1 )Û
−iĤ0 (t−t )
1
(t) =
i
τ2
−iĤ0 (t−t )
t
dt
t0
× Ĥ1 (t )e
(2.82)
(τ1 , t0 )
gives
dt e
(0)
t
Ĥ1 (t )e
dt e
−iĤ0 (t−t )
t0
−iĤ0 (t −t0 )
ψ (0) (t0 )
25
−iĤ0 (t −t0 )
ψ (0) (t0 )
Ĥ1 (t )e
−iĤ0 (t −t )
(2.83)
2 Quantum dynamics of simple systems
2.8 Adiabatic dynamics: Quantum adiabatic theorem
We consider a quantum system with discrete level structure and a Hamiltotian given by:
Ĥ(t) = Ĥ0 + V̂ (t).
(2.84)
The matrix representation of Ĥ in the basis of eigenstates of Ĥ0 is:
⎛
⎞
Vab (t)......
Ea
H(t) = ⎝ Vba (t) Eb ........ ⎠
.....
The solution of the corresponding TDSE can be written as
⎞
⎛
ψa (t)
⎜ ψb (t) ⎟
⎟
⎜
⎟
ψ(t) = ⎜
⎜ . ⎟
⎝ . ⎠
.
.
If the off-diagonal couplings Vij (t) are slowly varying one can define a unitary transformation that diagonalizes the instantaneous Hamiltonian H(t).
Where D(t) is a diagonal matrix and:
U −1 (t)H(t)U (t) = D(t)
U −1 ψ = ψ (2.85)
The TDSE can be written as
∂
Û
(t)
∂
∂ψ
+
ψ
i (Û ψ (t)) = i Û (t)
∂t
∂t
∂t
= Ĥ(t)Û (t)ψ (t)
i
(2.86)
∂ ∂ Û (t) ψ (t) = D̂(t)ψ (t) − iÛ −1 (t)
ψ (t)
∂t
∂t
If Ĥ(t) is slowly varying, so will be Û (t), and therefore also Û −1 (t). In this case
may be neglected. This is called the adiabatic approximation.
∂ Û (t)
∂t
Adiabatic theorem: If ψ(0) is an eigenfunction of Ĥ(0) and Ĥ(t) is a slowly-varying
function of time, then ψ(t) will evolve in such a way as to remain an eigenfunction of
Ĥ(t) for all time, in the limit of infinitely-slowly-varying Ĥ(t). (see A.Messiah Quantum Mechanics, North Holland Publishing Company, 1967 for the detailed derivation).
Applications:
1. The quantum adiabatic theorem can be used to calculate eigenfunctions for complicated potential as illustrated in Fig. 2.13.
2. It can be used to explain adiabatic alignment, i.e. the transition from field-free
rotational states to pendular states.
26
2 Quantum dynamics of simple systems
Figure 2.13: (a) Energy levels of the initial potential, V 0 (x ) (harmonic oscillator). (b)
Energy levels of the final potential V (x ) (double well). (c) Correlation
diagram of energy levels, E, as a function of a switching parameter, λ, that
changes the potential adiabatically from a harmonic oscillator to a double
well. An initial state that is an eigenstate of the oscillator will remain an
eigenstate of the switching potential and ultimately of the double well if the
switching time T → ∞ (taken from Introduction to quantum mechanics,
David J. Tannor)
2.9 Periodic Hamiltonians and Floquet theory
Consider a periodic Hamiltonian Ĥ(x, t + T ) = Ĥ(x, t) with period T. We use the ansatz
ψλ (x, t) = e−iλ t/ φλ (x, t).
(2.87)
Substituting into the TDSE and regrouping we find
ĤF (x, t)φλ (x, t) = λ φλ (x, t)
where
ĤF (x, t) = Ĥ(x, t) − i
∂
∂t
(2.88)
is the Floquet Hamiltonian
and φλ (x, t) is a Floquet eigenstate and λ is a Floquet energy (time independent).
Key result: Floquet eigenstates φλ (x, t) are periodic in time with the same
period T as the Hamiltonian.
The ansatz above is a particular solution whereas the general solution is given by
ψ(x, t) =
aλ e−iλ t/ φλ (x, t)
(2.89)
λ
27
2 Quantum dynamics of simple systems
2.10 Experimental decay in N-level problem
So far, we have considered oscillatory solutions of TDSE. In specific cases of relevance to
chemical kinetics, the early dynamics can display exponential character. In principle every isolated system with a time-independent Hamiltonian has only oscillatory
solutions. However, the oscillatory character may only appear after very long times e.g.
as revivals. We will now show that exponetial decays are a frequent early-time behavior
of auto-correlation functions in system with dense spectra.
The survival probability for a Lorentzian energy-level distribution is
Γ
1
2
p(E) ≈ pn (E) =
π (En − En,max )2 + ( Γ2 )2
(2.90)
One could consider a continuous spectrum or a dense spectrum of discrete levels
which we choose here. In this case, the time-dependent wavefunction is given by:
ψ(
r, t) =
∞
cn φn (
r)e
−iEn t
(2.91)
n=0
where |cn |2 = pn are the populations of the eigenstates φn (
r).
Figure 2.14: Lorentzian probability distribution in a dense spectrum
The ”Survival probabiltiy” of the initial state ψ(
r, t) = 0 is given by the auto-
28
2 Quantum dynamics of simple systems
correlation function:
pA (t) = |gA (t)|2
with gA (t) = ψ(
r, 0)|ψ(
r, t)
∞ −iEm t
(
c∗n φ∗n )(
cm φm e )d
r
=
−∞
=
n
|cn | e
2
m
−iEn t
(2.92)
n
=
pn e−iωn t .
n
For very dense spectra the sum can be replaced by an integral
∞
−iEt
gA (t) ≈
p(E)e dE
(2.93)
−∞
Γ
(2.94)
This is just a special case of the general relation between auto-correlation functions and
spectra.
−Γ
gA (t) ∝ e 2 t → pA (t) = e
−Γ
t
= e−kt with k =
Bixon-Jortner model
The derivation in the previous section 2.10 did not explain why one finds a Lorentzian
distribution of energy levels. This situation derives, among other cases, from the BixonJortner model and is frequently encountered in molecules.
We consider the model Hamiltonian:
Ĥ = Ĥ0 + V̂
and the matrix representation of the corresponding TISE:
29
(2.95)
2 Quantum dynamics of simple systems
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞⎛ a
n
Es0 V
V . . . .
(n)
⎜
0
⎟
V E1 0 0 . . . ⎟ ⎜ b1
⎜ (n)
V
0 E20 0 . . . ⎟
b2
⎟⎜
⎟⎜
.
⎜
⎟⎜ .
⎠⎝ .
.
.
.
⎛
⎞
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎟ = En ⎜
⎜
⎟
⎜
⎟
⎝
⎠
an
(n)
b1
(n)
b2
.
.
.
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
We describe the eigenfunctions and eigenvalues of Ĥ0 with ψs0 , Es0
The coupling matrix elements are given by V = ψk0 |Ĥ|ψs0 =const (real) for k=1,2,......
The eigenfunctions of Ĥ are
(n)
φn = an ψs0 +
bk ψk0
(2.96)
k
and the eigenvalues En are solutions of
Es0 − En −
π
πV 2
cot ( )(En − Es0 + α) = 0.
(2.97)
The eigenvalues En are nearly equidistant. One can show that each eigenvalue falls
between a pair of zero-order energies (ES0 ). One further finds that
|an |2 =
The limiting case
π2 V 2
2
V2
2
(En − Es0 )2 + V 2 + ( πV )2
(2.98)
(|V | >> ) leads to
>> 1
pn = |an |2 =
Γ
2
,
π (En − Es0 )2 + ( Γ2 )2
(2.99)
which defines a continuous population density
p(E) =
Γ
1
2
π (En − Es0 )2 + ( Γ2 )2
(2.100)
2
with half-width Γ = 2πV
.
Using a single zero-order state ψs as the initially populated state corresponds to a
typical situation where
∗one ”bright” level is excited.
One finds ψs =
n an φn (show this). Within the Bixon-Jortner model an is real,
hence cn = an , but the coefficients can also be complex. Since we now have a Lorentz
distribution pn = |an |2 of energies En for the functions φn , the survival probabiltiy is
exponential:
ps = pA (t) = |ψ(r, t)|ψs |2
= e−kt
with k =
Γ
h
=
2V 2
ρ
h2
where level density is ρ = 1 .
30
(2.101)
2 Quantum dynamics of simple systems
2.11 Derivation of Fermis’s golden rule for exponential decay
The so-called Fermi golden rule goes back to Wentzel and Dirac (∼1927) but was named
after Fermi’s lecture notes (∼1950)
Ĥ = Ĥ0 + V̂ (t)
We consider
Ĥ0 ψk (r) = Ek ψk (r)
−iEk t
and write
ψ(t) =
ck (t)φk e .
(2.102)
k
Here ck (t) are time-dependent because ψk are not eigenstates of Ĥ. Inserting ψ(t) into
TDSE yields (assuming V̂ to be time independent)
−iEk t
−iEk t
φk ċk (t)e =
e ck (t)V̂ φk
(2.103)
i
k
k
−iEk t
(V̂ acts only on φk and not on ck (t)e
).
Multiplying from the left with φ∗j and integrating over {r} yields:
iċj (t)e
−iEj t
=
e
−iEk t
ck (t)φj |V̂ |φk (2.104)
k
Multiplying with e
iEj t
yields
iċj (t) =
e−iωkj t ck (t)Vjk
(2.105)
k
Where ωkj =
Ek −Ej
and Vjk = φj |V̂ |φk or in matrix notation
i
where
d C = H̃C
dt
(2.106)
⎛
⎞
H̃11 (t) H̃12 (t)......
⎠
H̃ = ⎝ H̃21 (t)
........
.....
and
⎛
⎜
⎜
C=⎜
⎜
⎝
c1 (t)
c2 (t)
.
.
cn (t)
⎞
⎟
⎟
⎟
⎟
⎠
with H̃jk = eiωjk t Vjk where H̃ is time dependent. This can be avoided by applying the
transformation ak = e−iωk t ck , yielding
iȧj =
Vjk ak + ωj aj .
(2.107)
k
31
2 Quantum dynamics of simple systems
Using ωj = Ej and defining the diagonal matrix Ediag ≡ {Ej } one obtains
i
ȧ = {Ediag + V}
a = H(a)
a,
(2.108)
where H(a) is now time independent.
Both equations (2.106) and (2.108) look like the TDSE and can be solved numerically.
We now derive time dependence using perturbation theory.
The initial conditions are chosen to be: ck (0) = δkn , ψ(t = 0) = φn .
For short times: iċj = eiωjn t Vjn cn (0),
which is easy to integrate by separating the variables:
t
Vjn t iωjn t dcj =
e
dt
(2.109)
i 0
0
with Vjn time independent and otherwise:
t
1 t
dcj =
Vjn (t)eiωjn t dt
i 0
0
(2.110)
Equation (2.109) leads to:
cj (t) − cj (t = 0) = cj (t) =
pj (t) = |cj (t)|2 =
1
Vjn (eiωjn t − 1)
i2 ωjn
|Vjn |2 iωjn t
− 1|2
2 |e
2 ωjn
(2.111)
(2.112)
pj (t) represents the probability of a transition from the initial state n to a final state j
within the time interval (0,t). The matrix elements of the approximate time-evolution
operator are therefore given by
|Vjn |2
ωjn t 2
2
)
(2.113)
pj (t) = pjn (t) = |Ujn | = 2 2 4 sin(
2
ωjn
2
Exercise: Show that |eiα − 1|2 = 2 sin( α2 )
The transition probability is thus an oscillatory function of time within the short-time
ω t
ωjn t
approximation. Still for short times, one can write sin( jn
2 ) ≈ 2 , which inserted into
Equation (2.113) gives,
pjn (t) ≈ |Ujn |2 ≈
2 t2
|Vjn |2 t2
|Vjn |2 ωjn
=
4
2
4
2
2 ωjn
(2.114)
Note: Quadratic growth with time, no exponential time dependence!
32
2 Quantum dynamics of simple systems
Consider the sum of transition probabilities to many levels:
N
pjn
j=1
N
|Vjn |2
ωjn t 2
= 1 − pn (t) =
2 4 sin( 2 )
2 ωjn
j=1
(2.115)
This is the total probability, for all transitions leading away from an initial state n.
For dense levels, we replace the sum by an integral:
1 − pn (t) =
ΔE
2
−ΔE
2
|Vjn |2
ωjn t 2
)
4
sin(
p(Ej )dEj
2
2
2 ωjn
We write Ej = En + ωjn = En + 2x with x =
2
1 − pn (t) =
Δ
2
−Δ
2
2t2
1 − pn (t) =
|Vjn |
Δ
2
−Δ
2
2
ωjn
2
sin(xt)
x
(2.116)
and dEj = 2dx
2
ρ(En + 2x)dx
|Vjn | ρ(En + 2x)
2
sin(xt)
xt
(2.117)
2
dx
Using the approximation |Vjn |2 ρ(Ex ) ∼
= |V |2 ρ(E) and y = xt, it follows:
2t
1 − pn (t) = |V |2 ρ(E)
i.e t 2π
Δ
tΔ/2
sin2 y
dy
2
−tΔ/2 y
=π for tΔ>2π
(2.118)
but t must also remain small. Then
2π
|V |2 ρ(E)t
−dpn
2π
= kpn ≈ k =
|V |2 ρ
dt
1 − pn (t) =
(2.119)
Integration finally provides
pn (t) = e−kt
with k =
4π 2 |V |2 ρ
h
(2.120)
The result is the same as that obtained from the Bixon-Jortner model but with many
approximations.
33