More Scattering: the Partial Wave Expansion
Michael Fowler 1/17/08
Plane Waves and Partial Waves
We are considering the solution to Schrödinger’s equation for scattering of an incoming plane
wave in the
z
-direction by a potential localized in a region near the origin, so that the total wave
function beyond the range of the potential has the form
(
)
(
)
cos
,
,
,
.
ikr
ikr
e
r
e
f
r
θ
ψ
θ ϕ
θ ϕ
=
+
The overall normalization is of no concern, we are only interested in the
fraction
of the ingoing
wave that is scattered.
Clearly the outgoing current generated by scattering into a solid angle
at angle
θ
,
φ
is
d
Ω
(
)
2
,
f
d
θ ϕ
Ω
multiplied by a velocity factor that also appears in the
incoming wave.
Many potentials in nature are spherically symmetric, or nearly so, and from a theorist’s point of
view it would be nice if the experimentalists could exploit this symmetry by arranging to send in
spherical waves corresponding to different angular momenta rather than breaking the symmetry
by choosing a particular direction. Unfortunately, this is difficult to arrange, and we must be
satisfied with the remaining azimuthal symmetry of rotations about the ingoing beam direction.
In fact, though, a full analysis of the outgoing scattered waves from an ingoing plane wave
yields the same information as would spherical wave scattering.
This is because a plane wave
can actually be written as a
sum over
spherical
waves
:
.
cos
(2
1)
(
)
(cos
)
ik r
ikr
l
l
l
l
e
e
i
l
j
kr P
θ
θ
=
=
+
∑
G
G
Visualizing this plane wave flowing past the origin, it is clear that in spherical terms the plane
wave contains both incoming
and
outgoing spherical waves.
As we shall discuss in more detail
in the next few pages, the real function
(
)
l
j
kr
is a standing wave, made up of incoming and
outgoing waves of equal amplitude.
We are, obviously, interested only in the outgoing spherical waves
that originate by scattering
from the potential
, so we must be careful not to confuse the pre-existing outgoing wave
components of the plane wave with the
new
outgoing waves generated by the potential.
The radial functions
(
)
l
j
kr
appearing in the above expansion of a plane wave in its spherical
components are the
spherical Bessel functions
, discussed below.
The azimuthal rotational
symmetry of plane wave + spherical potential around the direction of the ingoing wave ensures
that the angular dependence of the wave function is just
(cos
)
l
P
θ
, not
(
)
,
lm
Y
θ ϕ
.
The coefficient

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