Ph235-08
F. Merritt
29-Oct-2008
Lecture 18 (November 7, 2008):
(version 1.0; 7-Nov-2008::00:00)
(Note: the first section of these notes repeats the last section of the notes from the last
lecture.)
Time Dependent Perturbation Theory: An Introduction
In Ph234, we applied separation of variables to the Schrodinger Equation
d
i
  r , t   H  r   r , t 
dt
and found that the general solution to the SE could be written as
  r , t    En  r   eiEnt /
(18.1)
(18.2)
where  En  r  is the eigenfunctions of H with eigenvalue En . Equivalently, we can
write the wavefunction as a Fourier sum
With the introduction of Dirac notation and Hilbert space, we derived the
propagator:
  r , t    U (r , t ; r )   r , 0 
(18.3)
where
or equivalently
 E 
U (r , t; r )   r En exp  i n t  En r 


n
(18.4)
(t )   En eiEnt / En (0)
(18.5)
n
where En are the complete set of eigenfunctions of the Hamiltonian H :
(18.6)
H En  En En
We have dealt almost exclusively with Hamiltonians of the form
P2
(18.7)
H
V r 
2M
where H has no explicit time dependence.
In all of this, the time dependence of the wavefunction was always governed by
the time dependence of the eigenfunctions. Although we have often dealt with
wavefunctions which change as a function of time (e.g., traveling wave packets and
group velocity), this time dependence has always been due to the changing phase and
interference of eigenstates. When the wavefunction is expanded in terms of the
eigenstates, the coefficients cn of the eigenstates En have not changed with time.
But this leaves out an enormous number of problems, including all those where
the Hamiltonian has explicit time dependence. For example, suppose a charged particle
moving at high velocity passes close to a hydrogen atom in its ground state. Then the
atom will see an electric field that varies as a function of time, and may become quite
large if the traveling particle passes close to the atom. In that case the atom may be
excited into a higher energy state, or may even be ionized. This clearly involves a
transition of the atom from one eigenstate to another, but nothing in the formalism we’ve
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developed so far allows us to compute the probability of this transition, because a) it
involves a transition from one eigenstate to another, and b) it involves a potential V  r 
that changes with time (i.e., V  r , t  ) due to the changing electric field of the moving
particle. Let’s call this “Example C” (C for charged particle).
Another example: we know that atoms can exist in excited states, and we even
know that a hydrogen atom can be excited from the ground state to the first excited state
by absorbing a photon of energy E2  E1 . But what is the actual mechanism and rate for
this? Does the photon have to have exactly the right energy? If so, this must be a very
rare occurrence – but we know that it is not at all rare. Let’s call this stimulated
transition phenomenon “Example B”.
Finally, and perhaps most perplexing, what happens to an atom in an excited
state? We know that it can decay to the ground state with the emission of a photon of
energy En  E1 , but what is the time constant? In fact, how can this happen at all? The
electron is presumably in an energy eigenstate En , and according to everything we have
learned (and equation (18.2) above) this should be a stable “stationary” state! It is an
eigenstate of the Hamiltonian. How can it decay? Let’s call this “Example A”.
Time Dependent Perturbation Theory addresses all of these examples. We will
approach it by extending our earlier development of perturbation theory to include the
case where the perturbation Hamiltonian H  is allowed to vary with time. In particular,
we will consider a Hamiltonian
(18.8)
H  H 0   H 1 (t )
where H 0 is a time-independent Hamiltonian with known eigenfunctions n0 , and
H 1 (t ) is a small time-dependent perturbation. [I have introduced the dimensionless
multiplier  in equation (18.8) in order to help us keep track of the order of the
approximation in what follows, just as we did in time-independent perturbation theory.
But it is not a critical part of the development, and most of the time I will not explicitly
write it. Just remember that it multiplies H 1 (t ) .]
Time Dependent Perturbation Theory
 t 
Picking up where we left off last lecture, we want to find how the wavefunction
develops under a Hamiltonian
H  H 0   H  t 
0
where H defines a complete set of eigenstates n
0
(18.9)
which satisfy the eigenvalue
equations
H 0 n0  En0 n0
(18.10)
From the Schrodinger equation, we know that
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d
 (t )   H 0  H 1  t    (t )
(18.11)
dt
Since any wavefunction can be expanded in terms of the complete set of basis states n0 ,
i
we can write
 En0t  0
(t )   cn  t  exp  
 n


(18.12)
where we have separated out the exponential time-dependence associated with the
eigenfunctions n0 . [Note that my cn  t  is the same as Shankar’s dn  t  ; I do not like
to have a variable called d in a differential equation, especially when we need to take
derivatives and differentials of this variable.] Substituting (18.12) into (18.11) gives
 En0t  0
 dcn
0
i

c
E
exp
n  dt n n   i  n


 En0t  0
 dcn 
i
exp
n  dt   i  n


 En0t 
0
0
1
0


H
n

H
t
n
c
exp


n 


 n


(18.13)
0
 En t 
1
0
   H  t  n  cn exp  

n



Left-multiplying by f 0 gives finally
dc f
dt
 
i
 
n
where
f 0 H 1  t  n0  cn  t  exp  i fn t 
 fn   E f  En  /
(18.14)
(18.15)
These equations are exact, following directly from the Schrodinger equation.
If the system starts out in an eigenstate i 0 , then cn  0   ni . In that case we can
approximate ci  t  by ci (0)  1 in equation (18.14), and integrate to get a first-order
calculation of c f (t ) :
c f  t    fi 
i
t

f 0 H 1  t   i 0 exp  i fn t   dt 
(18.16)
0
This is a very useful result, and we can immediately use it to solve problems. It’s
important to go through a few examples to understand exactly how this is applied.
Shankar uses it (p. 475-6) to find the probability that a SHO in the ground state will make
a transition to any excited state when it is subjected to the perturbation
H 1  t   eX exp  t 2 /  2  . The SHO provides a great example for this kind of
problem, because writing the operator X as a sum of raising and lowering operators
makes it easy to evaluate the matrix elements without doing integrals. The first problem
in HW6 provides a similar example, and the solution to Shankar 18.2.2 (which involves
the hydrogen atom and the application of selection rules) is given as an example on the
Ph235 web pages (week 6). All of these problems belong to the “Example C” category
of the “Introduction”.
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29-Oct-2008
Special types of time-dependent perturbation:
Sudden, Adiabation, and Periodic
There are three special cases of time-dependent perturbation which have rather
simple solutions, and also have application to a wide variety of real problems. It’s
important for the student to understand these, because they help a lot in developing both
intuition and mathematical competence. I will go through the first two here. The third
type, periodic perturbations, is the most important both because of its many applications,
and also because it is the starting point for future developments in field theory, QED,
nuclear and particle physics.
The Sudden Perturbation
We first encountered the “sudden perturbation” in the 3rd week of Ph234, when
we considered a ISW whose length L was instantaneously doubled to 2L. From the SE
alone, without using perturbation theory at all, we can see that the wavefunction cannot
change instantaneously. Its time derivative is proportional to H  , so an instantaneous
change in H produces an instantaneous change in d  / dt but not in  .
Another good example of the “sudden” perturbation is given by nuclear  -decay,
where a neutron in the nucleus decays by n  p  e  e . The electron leaves the
nucleus quickly, and the orbital electrons see a sudden change in the nuclear charge.
How does this affect the occupancy of the different energy levels?
In both cases, we can argue that the wavefunction cannot change quickly, so we
simply expand the wavefunction at t=0 (when the perturbation occurs) in terms of the
final eigenstates f m of the new Hamiltonian:
i   cm f m
where cm  f m i
(18.17)
m
If the change were really instantaneous, this would be the final result, since it is
expressed in term of eigenstates of the final Hamiltonian H f . But if the change takes
place over a time T, then
t
H (t )  H i   H f  H i 
T
(18.18)
t

 H f    1  H f  H i 
T 
Then the perturbation Hamiltonian is
t

H 1  t     1  H f  H i 
for 0  t  T
(18.19)
T

and the change in the cm ’s with respect to (18.17) is given by
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cn (T )  cn  0   
i
T

f n H 1  t  f m cm  0  exp  inm t  dt
m 0

i
T

m 0
t

f n   1  H f  H i  f m cm  0  exp  inm t  dt
T 
(18.20)
T  t


c
0
f
H

H
f
 1 exp  inm t  dt 


m m
n 
f
i
m  
0  T 

cx
cx
2
Using  xe dx  e  cx  1 / c , the integral over time becomes

i
T
T  t
  1  einmt  inmt  1 Teinmt  


    1 exp  inm t  dt    
 
2
nm
inm  0 
0  T 
  T 

T
 1
  2 einmt  inmt  1  inmTeinmt  
0
 nmT

(18.21)
 1

  2 einmT   1  inmT   
 nmT

nd
Expanding the exponential to 2 order (which is lowest non-vanishing order) we have
T  t
 T

(18.22)
   1 exp  inm t  dt  
0  T 
 2
Substituting this back into (18.20), we find that nm has vanished from the equation, and
that allows us to further simply the matrix element using (18.17):
i
 T 
cn (T )  cn  0    cn  0  En   cm  0  f n H i f m    
m

2
iT

(18.23)
cn  0  En   cm  0  f n H i f m  

2
m

iT

cn  0  En  f n H i  i 
2
If the system was in an eigenstate Ei of H i at t  0 , then Hi i  Ei i and the
equation becomes
iT
cn (T )  cn  0   
cn  0  En  Ei f n  i 
2
(18.24)
iT

cn  0   En  Ei 
2
Consequently the additional fractional change to cn caused by the non-zero time T is
small provided that

T
En  Ei
2
1
 2 n  i
1
(18.25)
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So we now know what “sudden” means quantitatively in the “sudden perturbation”, and
also have an approach which allows us to make higher order corrections if needed.
The Adiabatic Perturbation
An adiabatic change is just the opposite: a slow change, for example if we slowly
expand an ISW from length L to length 2L. In that case, a particle in the ground state of
the initial Hamiltonian will be in the ground state of the final Hamiltonian, provided the
time is much greater than the internal time constants i of the system.
It is possible to prove this in much the same way that we analyzed the “sudden
perturbation”, but I will not pursue this. A proof is given in Griffiths, pages 371-3.
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