MODULE 17
Time-dependent Perturbation Theory
In order to understand the properties of molecules it is necessary to see how systems respond to
newly imposed perturbations on their way to settling into different stationary states. This is
particularly true for molecules that are exposed to electromagnetic radiation where the
perturbation is a rapidly oscillating electromagnetic field. Time dependent perturbation theory
allows us to calculate transition probabilities and rates, very important properties in the
Photosciences.
The approach we shall use is that for the time-independent situation, viz., examine a two-state
system in detail and then outline the general case of arbitrary complexity.
The two-state system
We can write the total hamiltonian for the system as the sum of a stationary state term and a
time-dependent perturbation
Hˆ  Hˆ (0)  Hˆ (1) (t )
and suppose that the perturbation is one that oscillates at an angular frequency, , then
Hˆ (1) (t )  2 Hˆ (1) cos  t
(17.1)
(17.2)
where Ĥ (1) is the hamiltonian for a time-independent perturbation and the factor 2 is for later
convenience. Our time-dependent Schrödinger equation is
Ĥ   i

t
(17.3)
As before we consider a pair of eigenstates 1 and 2 representing the (time-independent)
wavefunctions  1 and  2 with energy eigenvalues E1 and E2. These wavefunctions are the
solutions of the equation
Hˆ (0) n  En n
(17.4)
and are related to the time-dependent functions according to
 n (t )   n e  iEnt /
1
(17.5)
When the perturbation is acting the state of the system can be expressed as a linear combination
of the basis functions defined by the set in equation (17.5). We confine ourselves to just two of
them and the linear combination becomes
(t )  a1 (t )1 (t )  a2 (t )2 (t )
(17.6)
Notice that we are allowing that the coefficients are time-dependent because the composition of
the state may evolve with time. The overall time dependence of the wavefunction is partly due
to the changes in the basis functions and partly due to the way the coefficients change with time.
2
At any time t, the probability that the system is in state n is given by an (t ) . Substitution of
equation (17.6) into the Schrödinger equation (17.3) leads to the following
Hˆ   a1 Hˆ (0) 1  a1 Hˆ (1) (t )1  a2 Hˆ (0)  2  a2 Hˆ (1) (t ) 2

(a11  a2  2 )
t
 1
a
 2
a
 i a1
 i 1 1  i a2
 i 2 2
t
t
t
t
i
(17.7)
Now each of the basis functions satisfies the time dependent Schrödinger equation, viz.
 n
Hˆ (0)  n  i
t
(17.8)
so the last equation in (17.7) simplifies down as
a1 Hˆ (1) (t )1  a2 Hˆ (1) (t ) 2  i a11  i a2  2
(17.9)
where the dots over the coefficients signify their time derivatives. Now we expand the last
equation by explicitly writing the time dependence of the wavefunctions
a1Hˆ (1) (t ) 1 e iE1t /  a2 Hˆ (1) (t ) 2 e iE2t /
 i a1 1 eiE1t /  i a2 2 e iE2t /
(17.10)
Now multiplying from the left by the bra 1 and use the orthonormality relationship we find
a1H11(1) (t )eiE1t /  a2 H12(1) (t )eiE2t /  i a1eiE1t /
(17.11)
where the matrix elements are defined in the usual way. Now we write 21  E2  E1 when we
find that (17.11) becomes
a1H11(1) (t )  a2 H12(1) (t )ei21t  i a1
2
(17.12)
It is not unusual to find that the perturbation has no diagonal elements so we put
(1)
H11(1) (t )  H 22
(t )  0
And then equation (17.12) reduces to
a2 H12(1) (t )ei21t  i a1
(17.13)
or, rearranging
a1 
1
a2 H12(1) (t )e  i21t
i
(17.14)
this provides us with a first order differential equation for one of the coefficients but it contains
the other one. We proceed exactly as above to obtain an equation for the other coefficient
a2 
1
(1)
a1 H 21
(t )ei21t
i
(17.15)
In the absence of any perturbation, the two matrix elements are both equal to zero and the time
derivatives of the two coefficients also become zero. In which case the coefficients retain their
initial values and even though  oscillates with time, the system remains frozen in its initially
prepared state.
Now suppose that a constant perturbation, V, is applied at some starting time and continued until
a time later. To allow this let us write
(1)
H12(1) (t )  V and H 21
(t )  V *
where we have invoked the Hermiticity of the hamiltonian.
a1  iVa2ei21t
Then
a2  iV * a1ei21t
(17.16)
This is a pair of coupled differential equations, which can be solved by one of several methods.
One solution is
a2 (t )  ( Aeit  Beit )ei21t / 2
(17.17)
where A and B are constants that are determined by the initial conditions and  is given by
1 2
2
  ( 21
 4 V )1/ 2
2
A similar expression holds fora1. 
3
(17.18)
If, prior to the perturbation being switched on the system is definitely in state 1 then a1(0) = 1,
and a2(0) = 0. These initial conditions allow us to find A and B and eventually we can arrive at
two particular solutions, one for a1(t) and one for a2(t). These are
i

  i21t
a1 (t )  cos t  21 sin t  e 2
2


1
(17.19)
and
iV
a2 (t )   
 
1
i 21t

2
sin

te


(17.20)
The probability of finding the system in state 2 (initially equal to zero) after turning on the
perturbation is given by
P2 (t )  a2 (t )
2
(17.21)
and by substitution of equation (17.20) we are lead to the Rabi formula
 4V 2
P2 (t )   2
  4 V
 21
 2 1 2
2 1/ 2 

sin
(


4
V
) t

21
2

2



(17.22)
Now suppose the pair of states is degenerate, then 21 = 0 and
P2 (t )  sin 2 V t
(17.23)
This function has the form shown in Figure 17.1.
1
0.8
g( t )
0.6
f ( t)
k( t )
0.4
0.2
0
0
2
4
6
8
10
t
Figure 17.1: three plots of the function in equation (17.23), with V = 1 (g), 2 (f), and 3(k)
4
The system oscillates between the two states, spending as much time in 1 as in 2. We also see
that the frequency of the oscillation depends on the strength of the perturbation, V, so that strong
perturbations drive the system between the two states more rapidly than do weak perturbations.
Such systems are described as ‘loose’, because even weak perturbations can drive them
completely from one extreme to the other.
The opposite situation is different, i.e. where the states are far apart in energy in comparison to
2
 4 V and
the absolute value of V. Then  21
2
2
2V 
2 1
P2 (t )  
 sin  21t
2
  21 
(17.24)
Again we find oscillation but now the probability of finding state 2 is never higher than
2
4 V /  21
which is much less than unity. Here the probability of the perturbation driving the
2
system into state 2 is very small. Furthermore, notice that the oscillation frequency is governed
by the energy separation and is independent of the perturbation strength. The only role of the
perturbation is to govern the fraction of the system in state 2. The probability is higher at larger
perturbations.
Many level systems
For the many-level system we need to expand the wavefunction as a linear combination of all the
n-states contained in equation (17.4). This leads to very complicated equations when combined
with perturbations that are more complex than the time-independent one used above.
To
approach this we use the approximation technique called “variation of constants” and invented
by Dirac. The problem we are faced with is to find out how the linear combination varies with
time under the influence of realistic perturbations. We base the approximation on the condition
that the perturbation is weak and applied for such a short time that all the coefficients remain
close to their initial values.
Eventually it can be shown that
t
1
i fi t
a f (t )   H (1)
dt
fi (t )e
i 0
5
(17.25)
where af is the coefficient for the final state that was initially unoccupied. This approximation
ignores the possibility that the route between initial and final states is via other states, i.e. it
accounts only for the direct route. It is a first order theory because the perturbation is applied
once and only once.
Now we can use the expression in (17.25) to see how a system will behave when exposed to an
oscillatory perturbation, such as light. First we consider transitions between a pair of discreet
states i and f . Then we shall embed the final state into a continuum of states.
Suppose our perturbation oscillates with a frequency  = 2 and it is turned on at t = 0. Then
its hamiltonian has the form
Hˆ (1) (t )  2 H (1) cos  t  H (1) (ei t  e i t )
(17.26)
(Now you see the reason for the factor of 2 in the expansion.). Putting this expression into
equation (17.25) yields
a1 (t ) 
H (1)
fi
i
t
 (e
i t
0
 e  i t )e
i fi t
dt
 i ( fi  )t  1 ei ( fi  ) t  1 
H (1)
fi  e




i  i ( fi   ) i ( fi   ) 
where
(17.27)
fi = Ef – Ei. This is an obscure result but simplification is very straightforward. In
electronic spectroscopy and photoexcitation the frequencies  and fi involved are very high, of
the order of 1015 s-1. Thus the first term in the braces is exceedingly small, whereas the second
term, in which the denominator can approach zero, can be very large. It is therefore possible to
ignore the first term. Then the probability of finding the system in state f after a time t, when it
was initially completely in state i, becomes
4 H (1)
fi
2
Pf (t )  a f (t ) 
2
2
( fi   )
sin 2
2
1
( fi   )t
2
(17.28)
1
(t )  V we rewrite equation (17.28) as
Using the relationship employed earlier that H12
Pf (t ) 
4 V fi
2
2 1
sin
( fi   )t
( fi   ) 2
2
6
(17.29)
This expression looks very much like equation (17.24) that was developed for a static
perturbation on the two-level system. The difference being that fi is here replaced by (fi - )
which is termed the frequency offset. Equation (17.29) informs us that not only is the amplitude
of the transition probability dependent on the frequency offset, so is its time dependence. Both
these increase as  approaches fi and when the difference becomes zero, the transition
probability is at a maximum; at this point the radiation field and the system are in resonance. 
Evaluating the limit of equation (17.29) as the two frequencies approach each other we find that
2
lim Pf (t )  V fi t 2
(17.30)
  fi
and the probability of the transition occurring as a result of the perturbation increases
quadratically with time.
The shape of equation (17.27) and the time dependence of the
amplitude for four times are shown in Figure 17.2.

22
19.8
17.6
15.4
g( f  1)
g( f  2)
g( f  3)
13.2
11
g( f  4) 8.8
6.6
4.4
2.2
0
20
16
12
8
4
0
4
8
12
16
20
r f
Figure 17.2: Plots of equation (17.27) at four different times (1,2,3, and 4) showing the
quadratic dependence of the amplitude on time. The vertical scale is arbitrary.
7
Transitions to states in a continuum
In the previous section the transition was between a pair of discrete states; now we consider the
case where the final state is part of a continuum of states, close to each other in energy. We can
still use equation (17.29), to estimate the probability of promoting to one member of the
continuum, but now we need to integrate over all the transition probabilities that the perturbation
can induce in the system. We need to define a density of states, (which relates to the
number of states accessible as a result of the perturbation, then  ( E )dE is the number of final
states in the range E to E + dE that are able to be reached. Then the total transition probability is
given by
P(t ) 

Pf (t )  ( E )dE
(17.31)
range
Now we use equation (17.29) and the relation  fi  E / to arrive at the following
1
P(t ) 

4 V fi
2
sin
( E /   )t
2
range
2
( E /   )2
 ( E )dE
(17.32)
Now we set about simplifying this expression. First, we recognize that the quotient in the
integrand, which is equivalent to sin 2 x / x 2 is very sharply peaked when E /   , the radiation
frequency. This means that we can sensibly restrict ourselves to considering only those states
that have significant transition probabilities at fi.
Because of this we can evaluate the
density of states in the neighborhood of E fi   fi and treat it as a constant. The other effect of
this is that although the matrix elements V fi depend on E, we are concerned only with a narrow
range of energies contributing to the integral, that we can assume it is a constant. Under these
considerations equation (17.32) simplifies to
1
2
P(t )  V fi  ( E fi )

sin 2 ( E /   )t
4
range
2
( E /   )2
dE
(17.33)
To add one more approximation we convert the integral into a standard form by extending the
limits to infinity. Because of the shape of the function, the value of the integrand has virtually no
area outside the actual range, so this approximation introduces very little error. Now we simplify
8
again by setting x 
1
( E /   )t , which implies that dE  (2 / t )dx , and the probability
2
expression becomes

2
sin x
2 
P(t )    V fi  ( E fi )t 2 
dx
2
x
 t 

2
(17.34)
Using the standard form that
2

 sin x 
  x  dx  
then the required probability becomes
2
P(t )  2 t V fi  ( E fi )
(17.35)
If we now define the transition rate (kfi) as the rate of change of the probability that the system
arrives at an initially empty state, we find that
2
k fi  2 V fi  ( E fi )
(17.36)
This expression is known as the Fermi Golden Rule and it tells us that we can calculate the rate
of a transition if we know the square modulus of the transition matrix element between the two
states and the density of final states at the frequency of the transition. It is a very useful
expression and we shall find many examples of its use in the coming weeks.
9