Relating Scattering Amplitudes to Bound States
Michael Fowler, UVa. 1/17/08
Low Energy Approximations for the
S
Matrix
In this section, we examine the properties of the partial-wave scattering matrix
(
)
(
)
1
2
l
l
S
k
ikf
k
=
+
for
complex
values of the momentum variable
k
.
Of course, general complex values of
k
do not
correspond to physical scattering, but it turns out that the scattering of physical waves can often
be most simply understood in terms of dominant singularities in the complex
k
plane.
We begin with the complex
k
connection between (positive energy) scattering and (negative
energy) bound states.
The asymptotic form of the
l
= 0 partial wave function in a scattering
experiment is (from the previous lecture)
(
)
0
2
ikr
ikr
S
k e
i
e
k
r
r
+
−
⎛
⎞
−
⎜
⎟
⎝
⎠
.
An
l
= 0 bound state has asymptotic wave function
r
Ce
r
κ
−
where
C
is a normalization constant.
Notice that this resembles an “outgoing wave” with imaginary momentum
.
If we
analytically continue the
scattering
wave function from real
k
into the complex
k
-plane, we get
both
exponentially increasing and decreasing wave functions, making no physical sense. But
there is an exception to this general observation: if the scattering matrix
becomes
infinite
at some complex value of
k
, the exponentially decreasing term will be infinitely larger than the
exponentially increasing term.
In other words, we’ll only have a decreasing wave function—a
bound state.
We know that the energy of a bound state has to be real and negative, and is also
equal to
, so this can only happen for
k
pure imaginary
,
k
i
κ
=
(
)
0
S
k
2
2
/ 2
k
=
m
k
i
κ
=
.
Now, the existence of a low energy bound state means that the
S
matrix has a pole (on the
imaginary axis) close to the origin, so this will strongly affect low energy (near the origin, but
real
k
) scattering.
Let’s see how that works using the low-energy approximation discussed
previously.
Recall that the
l
= 0 partial wave amplitude
(
)
(
)
(
)
0
0
1
,
cot
f
k
k
k
δ
=
i
−

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