*Michael Fowler 1/17/08*

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

_{}

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 _{} 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*:

_{}

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 _{}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_{}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 _{}, not _{}. The coefficient _{}is derived in Landau and Lifshitz, §34, by comparing the
coefficient of _{} on the two sides of
the equation: as we shall see below, _{}does not appear in _{} for *l* greater than *n*, and _{}does not appear in _{} for *l* less than *n*, so the *combination* _{}only occurs once—in the *n*^{th}
term, and the coefficients on both sides of the equation can be matched. (To get the coefficient right, we must of
course specify the normalizations for the Bessel function—see below—and the Legendre
polynomial.)

The plane wave _{} is a trivial solution
of Schrödinger’s equation with zero potential, and therefore, since the _{} form a linearly
independent set, each term _{} in the plane wave
series must be itself a solution to the zero-potential Schrödinger’s
equation. It follows that_{}satisfies the zero-potential radial Schrödinger equation:

_{}

The standard substitution _{} yields

_{}

For the simple case *l* = 0 the two solutions are _{}. The corresponding
radial functions *R*_{0}(*r*) are (apart from overall constants)
the zeroth-order *Bessel* and *Neumann* functions respectively.

The standard normalization for the zeroth-order Bessel function is

_{}

and the zeroth-order Neumann function

_{}

Note that the Bessel function is the one well-behaved at the origin: it could be generated by integrating out from the origin with initial boundary conditions of value one, slope zero.

Here is a plot of _{}from *kr* = 0.1 to
20:

For nonzero *l*,
near the origin is _{} The well-behaved _{} solution is the Bessel
function, the singular function the Neumann function. The standard normalizations of these
functions are given below.

Here are_{}:

*Detailed Derivation of Bessel and Neumann Functions*

*This subsection is
just here for completeness. We use the
dimensionless variable **r =
kr. *

To find the higher *l*
solutions, we follow a clever trick given in Landau and Lifshitz (§33).

Factor out the _{} behavior near the
origin by writing

_{}

The function _{} satisfies

_{}

The trick is to differentiate this equation with respect to *r *:

_{}

Writing purely formally _{}, the equation becomes

_{}

But this *is* the
equation that _{}must obey! So we have
a *recursion formula *for generating *all *the _{} from the zeroth one: _{} and _{}, up to a
normalization constant fixed by convention.

In fact, the standard normalization is

_{}

Now

_{}

This is a sum of only *even*
powers of *r*. It is easily checked that operating on this
series with _{}can never generate any negative powers of *r*. It follows
that_{}written as a power series in *r*, has leading term proportional to *r ^{ l}*.
The coefficient of this leading term can be found by applying the
differential operator to the series for

_{}

This *r ^{ l}* behavior near the origin is the usual
well-behaved solution to Schrödinger’s equation in the region where the
centrifugal term dominates.

Note that the small *r*
behavior is *not* immediately evident
from the usual presentation of the _{}’s, written as a mix of powers and trigonometric functions,
for example

_{}

Turning now to the behavior of the _{}’s for *large* *r*, from

_{}

it is evident that the dominant term in the large *r* regime (the
one of order 1/*r*) is generated by
differentiating *only* the
trigonometric function at each step.
Each such differentiation can be seen to be equivalent to multiplying by
(-1)
and subtracting p/2 from the argument, so

_{}

These _{}, then, are the physical partial-wave solutions to the
Schrödinger equation with zero potential.
When a potential is turned on, the wave function near the origin is
still _{} (assuming, as we
always do, that the potential is negligible compared with the _{}term sufficiently close to the origin). The wave function beyond the range of the
potential can be found numerically in principle by integrating out from the
origin, and in fact will be like _{}above *except* that
there will be an extra phase factor, called the “phase shift” and denoted by *δ*) in the sine. The significance of this is that in the far
region, the wave function is a linear combination of the Bessel function *and* the Neumann function (the solution
to the zero-potential Schrödinger equation singular at the origin). It is therefore necessary to review the
Neumann functions as well.

As stated above, the *l*
= 0 Neumann function is

_{}

the minus sign being the standard convention.

An argument parallel to the one above for the Bessel functions establishes that the higher-order Neumann functions are given by:

_{}

Near the origin

_{}

and for large *r*

_{}

so a function of the form _{} asymptotically can be
written as a linear combination of Bessel and Neumann functions in that region.

Finally, the spherical *Hankel*
functions are just the combinations of Bessel and Neumann functions that look
like outgoing or incoming plane waves in the asymptotic region:

_{}

so for large *r*,

_{}

Let us imagine for a moment that we could just send in a
(time-independent) spherical wave, with *θ*
variation given by *P _{l}*(cos

_{}

If now the (spherically symmetric) potential is turned on, the
only possible change to this standing wave solution in the faraway region
(where the potential is zero) is a phase shift *δ*:

_{}

This is what we would find on integrating the Schrödinger equation out from nonsingular behavior at the origin.

But in practice, the ingoing wave is given, and *its phase cannot be affected by switching on
the potential*. Yet we must still
have the solution to the same Schrödinger equation, so to match with the result
above we multiply the whole partial wave function by the phase factor _{}. The result is to put
*twice* the phase change onto the
outgoing wave, so that when the potential is switched on the change in the
asymptotic wave function must be

_{}

This equation introduces the *scattering matrix*

_{}

which must lie on the unit circle |*S*| =1 to conserve probability—the outgoing current must equal the
ingoing current. If there is no
scattering, that is, zero phase shift, the scattering matrix is unity.

It should be noted that when the radial Schrödinger’s
equation is solved for a nonzero potential by integrating out from the origin,
with _{}initially, the real function thus generated differs from the
wave function given above by an *overall*
phase factor _{}.

We’re now ready to take the ingoing plane wave, break it
into its partial wave components corresponding to different angular momenta,
have the partial waves individually phase shifted by *l*-dependent phases, and add it all back together to get the
original plane wave plus the scattered wave.

We are only interested here in the wave function far away from the potential. In this region, the original plane wave is

_{}

Switching on the potential phase shifts factor the outgoing wave:

_{}

The actual *scattering*
by the potential is the *difference*
between these two terms. The complete wave
function in the far region (including the incoming plane wave) is therefore:

_{}

The *i ^{ l}*
factor cancelled the

It is perhaps worth mentioning that for scattering in just
one partial wave, the outgoing current is equal to the ingoing current, whether
there is a phase shift or not. So, if
switching on the potential does not affect the total current scattered in any
partial wave, how can it cause any scattering?
The point is that for an ingoing *plane
wave* with zero potential, the ingoing and outgoing components have the
right relative phase to add to a component of a plane wave—a tautology,
perhaps. But if an extra phase is
introduced into the outgoing wave *only*,
the ingoing + outgoing will *no longer
give a plane wave*—there will be an extra outgoing part proportional to _{}.

Recall that the scattering amplitude _{} was defined in terms
of the solution to Schrödinger’s equation having an ingoing plane wave by

_{}

We’re now ready to express the scattering amplitude in terms of the partial wave phase shifts (for a spherically symmetric potential, of course):

_{}

where

_{}

is called the *partial
wave scattering amplitude*, or just the *partial
wave amplitude*.

So the total scattering amplitude is the sum of these partial wave amplitudes:

_{}

The total scattering cross-section

_{}

and the normalization of the Legendre polynomials

_{}

gives

So the total cross-section is the sum of the cross-sections
for each *l* value. This does *not* mean, though, that the differential
cross-section for scattering *into a given
solid angle* is a sum over separate *l*
values—the different components interfere.
It is only when *all* angles are
integrated over that the orthogonality of the Legendre polynomials guarantees
that the cross-terms vanish.

Notice that the maximum possible scattering cross-section
for particles in angular momentum state *l*
is _{} which is *four times* the classical cross section
for that partial wave impinging on, say, a hard sphere! (Imagine semiclassically particles in an
annular area: angular momentum *L* = *rp*, say, but _{} and _{} so *l* = *rk. * Therefore the annular area corresponding to
angular momentum “between” *l* and *l* + 1 has inner and outer radii _{}and _{}and therefore area _{}.) The quantum result
is essentially a diffractive effect, we’ll discuss it more later.

It’s easy to prove the *optical
theorem* for a spherically-symmetric potential: just take the imaginary part
of each side of the equation

_{}

at *θ* = 0,
using _{}

_{}

from which the optical theorem _{} follows immediately.

It’s also worth noting what the unitarity of the *l*^{th} partial wave scattering
matrix_{} implies for the partial wave amplitude _{}. Since _{} it follows that

_{}

From this, _{}gives:

_{}

This can be put more simply:

_{}

In fact,

_{}

We assume in this section that the potential can be taken to
be zero beyond some boundary radius *b*. This is an adequate approximation for all
potentials found in practice *except*
the Coulomb potential, which will be discussed separately later.

Asymptotically, then,

_{}

This expression is only exact in the limit _{} but since the
potential can be taken zero beyond *r*
= *b*, the wave function must have the
form

_{}

for *r* > *b*.

(The - sign comes from the standard convention for Bessel and Neumann functions—see earlier.)

*The Hard Sphere*

The simplest example of a scattering potential:

_{}

The wave function must equal zero at *r* = *R*, so from the above
form of _{},

_{}

For* l* = 0,

_{}

so _{} This amounts to the wave
function being effectively moved over to begin at *R* instead of at the origin:

_{}

for *r* > *R*, of course _{} for *r*
< *R*.

For higher angular momentum states at low energies (*kR* << 1),

_{}

Therefore at low enough energy, only *l* =
0 scattering is important—as is obvious, since an incoming particle with
momentum _{}and angular momentum _{} is most likely at a
distance *l*/*k* from the center of the
potential at closest approach, so if this is much greater than *R*, the phase shift will be small.

From the definition of _{}

_{}

and

_{}

recall the Born approximation amounts to replacing the wave
function _{}in the integral on the right by the incoming plane wave,
therefore ignoring rescattering.

To translate this into a partial wave approximation, we first
take the incoming ** _{}** to be in the

_{}

Labeling the angle
between ** _{}** and

_{}

Now** _{}_{ }** is in the direction

_{}

On inserting this
expression and integrating over _{}, the nonzero *m*
terms give zero, in fact the only nonzero term is that with the same *l* as the term in the _{} expansion, giving

_{}

and remembering

_{}

it follows that for small phase shifts (the only place it’s valid) the partial-wave Born approximation reads

_{}

From

_{}

the *l* = 0 cross
section is

_{}

At energy _{} the radial Schrödinger equation for _{} away from the potential
becomes _{}, with a straight line solution _{} This must be the _{} limit of _{}, which can only be correct if *δ*_{0} is itself linear in *k* for sufficiently small *k*,
and then it must be _{} *a* being the point at which the extrapolated external wavefunction
intersects the axis (maybe at negative *r*!) So, as *k*
goes to zero, the cot term dominates in the denominator and

_{}

The quantity *a* is
called the *scattering length*.

Integrating the zero-energy radial Schrödinger equation out
from *u*(*r*) = 0 at the origin for a weak (spherical) square well potential,
it is easy to check that *a* is
positive for a repulsive potential, negative for an attractive potential.

Repulsive potential, zero-energy wave function (so it’s a *straight line* outside of the well!):

Attractive potential:

On increasing the strength of the repulsive potential, still
solving for the zero-energy wave function, *a*
tends to the potential wall—here’s the zero-energy wavefunction for a barrier
of height 6:

For an infinitely
high barrier, the wave function is pushed out of the barrier completely, and
the hard sphere result is recovered: scattering length *a*, cross-section 4*π* *a*^{2}.

On increasing the strength of the *attractive* well, if there is a phase change greater that *π*/2 within the well, *a* will become positive. In fact, right
at *π*/2, *a* is infinite!

And a little more depth to the well gives a *positive* scattering length:

In fact, a well deep enough to have a positive scattering
length will also have a bound state. This becomes evident when one considers
that the depth at which the scattering length becomes infinite can be thought
of as formally having a zero energy bound state, in that although the wave
function outside is not normalizable, it is equivalent to an exponentially
decaying function with infinite decay length.
If one now deepens the well a little, the zero-energy wave function
inside the well curves a little more rapidly, so the slope of the wave function
at the edge of the well becomes negative, as in the picture above. With this slightly deeper well, we can now *lower* the energy slightly to negative
values. This will have little effect on
the wave function inside the well, but make possible a fit at the well edge to
an exponential decay outside—a genuine bound state, with wave function _{} outside the well.

If the binding energy of the state is really low, the
zero-energy scattering wave function inside the well is almost identical to
that of this very low energy bound state, and in particular the logarithmic
derivative at the wall will be very close, so _{}, taking *a* to be
much larger than the radius of the well.

This connects the large scattering length to the energy of the weakly bound state,

_{}

(Sakurai, p 414.)

Wigner was the first to use this to estimate the binding energy of the deuteron from the observed cross section for low energy neutron-proton scattering.