Phys 852, Quantum mechanics II, Spring 2008
Introduction to Scattering Theory
Statement of the problem:
Scattering theory is essentially timeindependent perturbation theory applied to the case of a continuous
spectrum. That means that we know there is an eigenstate of the full Hamiltonian for every possible energy,
E
. Thus the job of finding the full eigenvalues, which was a major part of TIPR, is not necessary here.
In scattering theory, we just pick any
E
, and then try to find the ‘perturbed’ eigenstate,

ψ
(
E
)
)
. On the
other hand, remember that there are usually multiple degenerate eigenstates for any given energy.
So
the question becomes; which of the presumably infinitely many degenerate fulleigenstates are we trying
to compute? The answer comes from causality; we want to be able to completely specify the probability
current amplitude coming
in
from
vector
r
=
∞
, and then we want the theory to give us the corresponding
outgoing current amplitude.
The way we do this is to pick an ‘unperturbed’ eigenstate which has the
desired incoming current amplitude (we don’t need to worry what the outgoing current amplitude of
the unperturbed state is).
The second step is to make sure that our perturbation theory generates no
additional incoming currents, which we accomplish by putting this condition in by hand, under the mantra
of ‘causality’. As we will see, this means that the resulting ‘full eigenstate’ will have the desired incoming
current amplitude. Now if you go back to what you know, you will recall that ‘solving’ a partial differential
equation requires first specifying the desired boundary conditions, which is exactly what the standard
scattering theory formalism is designed to do.
Typically, the scattering formalism is described in the following way: an incident particle in state

ψ
0
)
is scattered by the potential
V
, resulting in a scattered state

ψ
s
)
. The incident state

ψ
0
)
is assumed to be
an eigenstate of the ‘background’ hamiltonian
H
0
, with eigenvalue
E
. This is expressed mathematically as
(
E
−
H
0
)

ψ
0
)
= 0
.
(1)
Unless otherwise specified, the background Hamiltonian should be taken as that of a freeparticle,
H
0
=
P
2
2
M
,
(2)
and the incident state taken as a plane wave
(
vector
r

ψ
0
)
=
ψ
0
(
vector
r
) =
e
i
vector
k
·
vector
r
.
(3)
As with onedimensional scattering, we do not need to worry about the normalization of the incident state.
Furthermore, the potential
V
(
vector
R
) is assumed to be ‘localized’, so that
lim
r
→∞
V
(
vector
r
) = 0
.
(4)
The goal of scattering theory is then to solve the full energyeigenstate problem
(
E
−
H
0
−
V
)

ψ
)
= 0
,
(5)
where
E >
0 (unless otherwise specified), and

ψ
)
is the eigenstate of the full Hamiltonian
H
=
H
0
+
V
with energy
E
. It should be clear that there is a different

ψ
0
)
and correspondingly, a different

ψ
)
for each
energy
E
, even though our notation does not indicate this explicitly.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Moore
 mechanics, SCATTERING, lim, KR, σtot, lim ψ

Click to edit the document details