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Chapter 12
Time-dependent perturbation
theory
So far, we have focused largely on the quantum mechanics of systems in
which the Hamiltonian is time-independent. In such cases, the time depen-
dence of a wavepacket can be developed through the time-evolution operator,
U = e
−i ˆ Ht/ℏ
or, when cast in terms of the eigenstates of the Hamiltonian,
H|n〉 = E n
|n〉, as |ψ(t)〉 = e
−i
ˆ Ht/ℏ |ψ(0)〉 =
n
e
−iEnt/ℏ c n
(0)|n〉. Although
this framework provides access to any closed quantum mechanical system, it
does not describe interaction with an external environment such as that im-
posed by an external electromagnetic field. In such cases, it is more convenient
to describe the induced interactions of a small isolated system,
H
0
, through a
time-dependent interaction V (t). Examples include the problem of magnetic
resonance describing the interaction of a quantum mechanical spin with an
external time-dependent magnetic field, or the response of an atom to an ex-
ternal electromagnetic field. In the following, we will develop a formalism to
treat time-dependent perturbations.
12.1 Time-dependent potentials: general formalism
Consider then the Hamiltonian
H =
H
0
- V (t), where all time-dependence
enters through the potential V (t). In the Schr¨odinger representation, the
dynamics of the system are specified by the time-dependent wavefunction,
|ψ(t)〉 S
through the Schr¨odinger equation iℏ∂ t
|ψ(t)〉 S
H|ψ(t)〉 S
. However,
in many cases, and in particular with the current application, it is convenient
to work in the Interaction representation,
1
defined by
|ψ(t)〉 I
= e
i
ˆ H 0 t/ℏ
|ψ(t)〉 S
where |ψ(0)〉 I
= |ψ(0)〉 S
. With this definition, one may show that the wave-
function obeys the equation of motion (exercise)
iℏ∂ t |ψ(t)〉 I
= V
I (t)|ψ(t)〉 I
where V I
(t) = e
i
ˆ H 0 t/ℏ
V e
−i
ˆ H 0 t/ℏ
. Then, if we form the eigenfunction expansion,
|ψ(t)〉 I
n
c n (t)|n〉, and contract the equation of motion with a general
state, 〈n|, we obtain
iℏ c˙ m
(t) =
n
V
mn
(t)e
iω mn t
c n
(t) , (12.2)
1
Note how this definition differs from that of the Heisenberg representation, |ψ〉 H =
e
i ˆ Ht/ℏ
|ψ(t)〉 S in which all time-dependence is transferred into the operators.
12.1. TIME-DEPENDENT POTENTIALS: GENERAL FORMALISM 140
where the matrix elements V mn
(t) = 〈m|V (t)|m〉, and ω mn
= (E
m
− E
n
−ω nm
. To develop some intuition for the action of a time-dependent potential,
it is useful to consider first a periodically-driven two-level system where the
dynamical equations can be solved exactly.
$ Info. The two-level system plays a special place in the modern development
of quantum theory. In particular, it provides a platform to encode the simplest
quantum logic gate, the qubit. A classical computer has a memory made up of
bits, where each bit holds either a one or a zero. A quantum computer maintains a
sequence of qubits. A single qubit can hold a one, a zero, or, crucially, any quantum
superposition of these. Moreover, a pair of qubits can be in any quantum superposition
of four states, and three qubits in any superposition of eight. In general a quantum
computer with n qubits can be in an arbitrary superposition of up to 2
n different
states simultaneously (this compares to a normal computer that can only be in one
of these 2
n states at any one time). A quantum computer operates by manipulating
those qubits with a fixed sequence of quantum logic gates. The sequence of gates to
be applied is called a quantum algorithm.
An example of an implementation of qubits for a quantum computer could start
with the use of particles with two spin states: |↓〉 and |↑〉, or | 0 〉 and | 1 〉). In fact any
system possessing an observable quantity A which is conserved under time evolution
and such that A has at least two discrete and sufficiently spaced consecutive eigenval-
ues, is a suitable candidate for implementing a qubit. This is true because any such
system can be mapped onto an effective spin-1/2 system.
$ Example: Dynamics of a driven two-level system: Let us consider a
two-state system with
H
0
E
1
0 E
2
, V (t) =
0 δe
iωt
δe
−iωt
0
Specifying the wavefunction by the two-component vector, c(t) = (c 1
(t) c 2
(t)), Eq. (12.2)
translates to the equation of motion (exercise)
iℏ∂ t
c = δ
0 e
i(ω−ω 21 )t
e
−i(ω−ω 21 )t 0
c(t) ,
where ω 21
= (E
2
− E
1 )/ℏ. With the initial condition c 1 (0) = 1, and c 2 (0) = 0, this
equation has the solution,
|c 2 (t)|
2
=
δ
2
δ
2
2 (ω − ω 21
2 / 4
sin
2
Ωt, |c 1 (t)|
2
= 1 − |c 2 (t)|
2
,
where Ω = ((δ/ℏ)
2 +(ω −ω 21
2 /4)
1 / 2 is known as the Rabi frequency. The solution,
which varies periodically in time, describes the transfer of probability from state 1 to
state 2 and back. The maximum probability of occupying state 2 is a Lorentzian with
|c 2 (t)|
2
max
γ
2
γ
2
2 (ω − ω 21
2 / 4
taking the value of unity at resonance, ω = ω 21
$ Exercise. Derive the solution from the equations of motion for c(t). Hint:
eliminate c 1 from the equations to obtain a second order differential equation for c 2
$ Info. The dynamics of the driven two-level system finds practical application
in the Ammonia maser: The ammonia molecule NH 3
has a pryramidal structure
with an orientation characterised by the position of the “lone-pair” of electrons sited
12.2. TIME-DEPENDENT PERTURBATION THEORY 142
where the term n = 0 translates to I. Note that the operators V I (t) are organised in
a time-ordered sequence, with t 0
≤ t n
≤ t n− 1
≤ · · · t 1
≤ t. With this understanding,
we can write this expression more compactly as
U
I
(t, t 0
) = T
[
e
−
i
ℏ
R t
t 0
dt
′ V I (t
′ )
]
where “T” denotes the time-ordering operator and its action is understood by Eq. (12.3).
If a system is prepared in an initial state, |i〉 at time t = t 0
, at a subsequent
time, t, the system will be in a final state,
|i, t 0
, t〉 = U I
(t, t 0
)|i〉 =
n
|n〉
c n (t)
〈n|U I
(t, t 0
)|i〉.
Making use of Eq. (12.3), and the resolution of identity,
m
|m〉〈m| = I, we
obtain
c n
(t) =
c
(0)
n
δ ni
c
(1)
n
i
t
t 0
dt
′
〈n|V I
(t
′
)|i〉
c
(2)
n
2
t
t 0
dt
′
t
′
t 0
dt
′′
m
〈n|V I
(t
′
)|m〉〈m|V I
(t
′′
)|i〉 + · · ·.
Recalling that V I = e
i
ˆ H 0 t/ℏ
V e
−i
ˆ H 0 t/ℏ
, we thus find that
c
(1)
n
(t) = −
i
t
t 0
dt
′
e
iω ni t
′
V
ni
(t
′
)
c
(2)
n
(t) = −
2
m
t
t 0
dt
′
t
′
t 0
dt
′′
e
iω nm t
′ +iω mi t
′′
V
nm
(t
′
)V mi
(t
′′
) ,
where V nm (t) = 〈n|V (t)|m〉 and ω nm
= (E
n
− E
m )/ℏ, etc. In particular, the
probability of effecting a transition from state |i〉 to state |n〉 for n &= i is given
by P i→n
= |c n
(t)|
2 = |c
(1)
n
(t) + c
(2)
n
(t) + · · · |
2 .
$ Example: The kicked oscillator: Suppose a simple harmonic oscillator is
prepared in its ground state | 0 〉 at time t = −∞. If it is perturbed by a small time-
dependent potential V (t) = −eEx e
−t
2 /τ
2
, what is the probability of finding it in the
first excited state, | 1 〉, at t = +∞?
Working to the first order of perturbation theory, the probability is given by
P
0 → 1
(1)
1
2 where c
(1)
1
(t) = −
i
ℏ
t
t 0
dt
′ e
iω 10 t
′
V
10
(t
′ ), V 10
(t
′ ) = −eE〈 1 |x| 0 〉e
−t
′ 2 /τ
2
and ω 10
= ω. Using the ladder operator formalism, with | 1 〉 = a
† | 0 〉 and x =
√
ℏ
2 mω
(a + a
†
), we have 〈 1 |x| 0 〉 =
ℏ
2 mω
. Therefore, making use of the identity
∞
−∞
dt
′ exp[iωt
′ − t
′
2
/τ
2 ] =
πτ exp[−ω
2 τ
2 /4], we obtain the transition amplitude,
c
(1)
1
(t → ∞) = ieEτ
π
2 mℏω
e
−ω
2 τ
2 / 4
. As a result, we obtain the transition probabil-
ity, P 0 → 1
2 (π/ 2 mℏω)e
−ω
2 τ
2 / 2
. Note that the probability is maximized for
τ ∼ 1 /ω.
$ Exercise. Considering the same perturbation, calculate the corresponding
transition probability from the ground state to the second excited state. Hint: note
that this calculation demands consideration of the second order of perturbation theory.
12.3. “SUDDEN” PERTURBATION 143
12.3 “Sudden” perturbation
To further explore the time-dependent perturbation theory, we turn now to
consider the action of fast or “sudden” perturbations. Here we define sudden
as a perturbation in which the switch from one time-independent Hamiltonian
H
0
to another
H
′
0
takes place over a time much shorter than any natural
period of the system. In this case, perturbation theory is irrelevant: if the
system is initially in an eigenstate |n〉 of
H
0
, its time evolution following the
switch will follow that of
H
′
0
, i.e. one simply has to expand the initial state as
a sum over the eigenstates of
H
′
0
, |n〉 =
n
′ |n
′
〉〈n
′
|n〉. The non-trivial part
of the problem lies in establishing that the change is sudden enough. This is
achieved by estimating the actual time taken for the Hamiltonian to change,
and the periods of motion associated with the state |n〉 and with its transitions
to neighboring states.
12.3.1 Harmonic perturbations: Fermi’s Golden Rule
Let us then consider a system prepared in an initial state |i〉 and perturbed by
a periodic harmonic potential V (t) = V e
−iωt
which is abruptly switched on at
time t = 0. This could represent an atom perturbed by an external oscillating
electric field, such as an incident light wave. What is the probability that, at
some later time t, the system lies in state |f〉?
From Eq. (12.4), to first order in perturbation theory, we have
c
(1)
f
(t) = −
i
t
0
dt
′
〈f|V |i〉e
i(ω fi −ω)t
′
i
〈f|V |i〉
e
i(ω fi −ω)t
− 1
i(ω fi
− ω)
The probability of effecting the transition after a time t is therefore given by
Plot of sin
2
(αt)/α
2 for t = 1.
Note that, as t → ∞, this func-
tion asymptotes to a δ-function,
πtδ(α).
P
i→f (t) * |c
(1)
f
(t)|
2
2
|〈f|V |i〉|
2
sin((ω fl − ω)t/2)
(ω fl
− ω)/ 2
2
Setting α = (ω fl
− ω)/2, the probability takes the form sin
2
(αt)/α
2 with a
peak at α = 0, with maximum value t
2 and width of order 1/t giving a total
weight of order t. The function has more peaks positioned at αt = (n + 1/2)π.
These are bounded by the denominator at 1/α
2
. For large t their contribution
comes from a range of order 1/t also, and as t → ∞ the function tends towards
a δ-function centred at the origin, but multiplied by t, i.e. the likelihood of
transition is proportional to time elapsed. We should therefore divide by t to
get the transition rate.
Finally, with the normalisation,
∞
−∞
dα(
sin(αt)
α
2 = πt, we may effect the
replacement, lim t→∞
1
t
sin(αt)
α
2
= πδ(α) = 2πδ(2α) leading to the following
expression for the transition rate,
Enrico Fermi 1901-1954:
An Italian physi-
cist most noted
for his work on
the development
of the first
nuclear reactor,
and for his
contributions to
the development
of quantum
theory, nuclear and particle physics,
and statistical mechanics. Fermi was
awarded the Nobel Prize in Physics
in 1938 for his work on induced
radioactivity and is today regarded
as one of the most influential
scientists of the 20th century. He is
acknowledged as a unique physicist
who was highly accomplished in both
theory and experiment. Fermium, a
synthetic element created in 1952 is
named after him.
R
i→f (t) = lim
t→∞
P
i→f (t)
t
2 π
2
|〈f|V |i〉|
2
δ(ω fl − ω). (12.5)
This expression is known as Fermi’s Golden Rule.
2
One might worry that,
in the long time limit, we found that the probability of transition is in fact
2
Curiously, although named after Fermi, most of the work leading to the Golden Rule was
undertaken in an earlier work by Dirac, (P. A. M. Dirac, The quantum theory of emission and
absorption of radiation. Proc. Roy. Soc. (London) A 114 , 243265 (1927)) who formulated
an almost identical equation, including the three components of a constant, the matrix
element of the perturbation and an energy difference. It is given its name due to the fact
that, being such a useful relation, Fermi himself called it “Golden Rule No. 2” (E. Fermi,
Nuclear Physics, University of Chicago Press, 1950).
12.3. “SUDDEN” PERTURBATION 145
This is a transition in which the system gains energy 2ℏω from the harmonic
perturbation, i.e. two “photons” are absorbed in the transition, the first taking
the system to the intermediate energy ω m , which is short-lived and therefore
not well defined in energy – indeed there is no energy conservation requirement
for the virtual transition into this state, only between initial and final states.
Of course, if an atom in an arbitrary state is exposed to monochromatic light,
other second order processes in which two photons are emitted, or one is
absorbed and one emitted (in either order) are also possible.
