Ph235-08
F. Merritt
29-Oct-2008
Lecture 17 (November 5, 2008):
(version 1.0; 24-Oct-2008::00:00)
Much of the lecture was spent going over some of the midterm problems:
numbers 4, 5, and 6. All of these involve degenerate perturbation theory. Solutions are
posted.
The midterm average was 41.3 out of 100. This is not disastrous, but I think it
should have been better. For this reason, I’m assigning all of the midterm problems as
part of HW6 (but if you got a score of 16/20 or better for any problem, you can skip that
one if you like).
The last part of the lecture was devoted to an introduction to Time Dependent
Perturbation Theory.
Time-Dependent Perturbation Theory: 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 /
(0.1)
(0.2)
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 
(0.3)
where
or equivalently
 E 
U (r , t ; r )   r En exp  i n t  En r 


n
(0.4)
(t )   En eiEnt / En (0)
(0.5)
n
where En are the complete set of eigenfunctions of the Hamiltonian H :
(0.6)
H En  En En
We have dealt almost exclusively with Hamiltonians of the form
P2
H
 V  r 
(0.7)
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
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Ph235-08
F. Merritt
29-Oct-2008
group velocity), this time dependence has always been due to the interference of
eigenstates.
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 b e
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
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 (0.2) 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
introduce 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
H  H 0   H 1 (t )
(0.8)
where H 0 is a time-independent Hamiltonian with known eigenfunctions n 0 , and
H 1 (t ) is a small time-dependent perturbation. [I have introduced the dimensionless
multiplier  in equation (0.8) 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 ) .]
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Ph235-08
F. Merritt
29-Oct-2008
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