Scattering: More the Partial Wave Expansion
Michael Fowler 1/17/08
Plane Waves and Partial Waves
We are considering the solution to Schrdingers 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
( r , , ) = eikr cos + f ( , )
eikr
.
r
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
d at angle , is f ( , ) d multiplied by a velocity factor that also appears in the
2
incoming wave.
Many potentials in nature are spherically symmetric, or nearly so, and from a theorists 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:
eik .r = eikr cos = i l (2l + 1) jl ( kr ) Pl (cos )
l
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 jl (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 jl (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 Pl (cos ) , not Ylm ( , ) . The coefficient
2
i l ( 2l + 1) is derived in Landau and Lifshitz, 34, by comparing the coefficient of ( kr cos ) on
n
the two sides of the equation: as we shall see below, ( kr ) does not appear in jl ( kr ) for l greater
n
than n, and ( cos ) does not appear in Pl (cos ) for l less than n, so the combination
n
( kr cos )
n
only occurs oncein the nth 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 functionsee belowand the Legendre polynomial.)
Mathematical Interval: The Spherical Bessel and Neumann Functions
The plane wave eik .r is a trivial solution of Schrdingers equation with zero potential, and
therefore, since the Pl (cos ) form a linearly independent set, each term jl (kr ) Pl ( cos ) in the
plane wave series must be itself a solution to the zero-potential Schrdingers equation. It
follows that jl (kr ) satisfies the zero-potential radial Schrdinger equation:
l ( l + 1)
d2
2d
Rl ( r ) +
Rl ( r ) + k 2
Rl ( r ) = 0.
2
dr
r dr
r2
The standard substitution Rl ( r ) = ul ( r ) / r yields
d 2ul ( r ) 2 l ( l + 1)
+k
u (r ) = 0
dr 2
r2
For the simple case l = 0 the two solutions are u0 ( r ) = sin kr , cos kr . The corresponding radial
functions R0(r) are (apart from overall constants) the zeroth-order Bessel and Neumann functions
respectively.
The standard normalization for the zeroth-order Bessel function is
j0 ( kr ) =
sin kr
,
kr
and the zeroth-order Neumann function
n0 ( kr ) =
cos kr
.
kr
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.
3
Here is a plot of j0 ( kr ) and n0 (kr ) from kr = 0.1 to 20:
For nonzero l, near the origin is Rl (r ) r l or r ( ) . The well-behaved r l solution is the
Bessel function, the singular function the Neumann function. The standard normalizations of
these functions are given below.
l +1
Here are j5 (kr ) and j50 (kr ) :
4
Detailed Derivation of Bessel and Neumann Functions
This subsection is just here for completeness. We use the dimensionless variable = kr.
To find the higher l solutions, we follow a clever trick given in Landau and Lifshitz (33).
Factor out the l behavior near the origin by writing
5
Rl = ( ) l ( ) .
l
The function l ( ) satisfies
2 ( l + 1) d
d2
l ( ) +
( ) + l ( ) = 0.
2
d l
d
The trick is to differentiate this equation with respect to :
2 ( l + 1) d 2
2 ( l + 1) d
d3
l ( ) +
l ( ) + 1
( ) = 0.
3
2
d
2 d l
d
Writing purely formally
d
l ( ) = l +1 ( ) , the equation becomes
d
2 (l + 2) d
d2
l +1 ( ) +
( ) + l +1 ( ) = 0.
2
d l +1
d
But this is the equation that l +1 ( ) must obey! So we have a recursion formula for generating
all the jl ( ) from the zeroth one: l +1 ( ) =
1d
l
l ( ) , and jl ( ) = ( ) l ( ) , up to a
d
normalization constant fixed by convention.
In fact, the standard normalization is
l
1 d sin
jl ( ) = ( )
.
d
l
Now
n
( sin ) / = ( 1) 2 n / ( 2n + 1)!
0
This is a sum of only even powers of . It is easily checked that operating on this series with
l
1 d
can never generate any negative powers of . It follows that jl ( ) , written as a power
d
series in , has leading term proportional to l. The coefficient of this leading term can be
found by applying the differential operator to the series for ( sin ) / ,
6
jl ( )
l
( 2l + 1)!!
as 0.
This r l behavior near the origin is the usual well-behaved solution to Schrdingers equation in
the region where the centrifugal term dominates.
Note that the small behavior is not immediately evident from the usual presentation of the
jl ( ) s, written as a mix of powers and trigonometric functions, for example
j1 ( ) =
sin
2
cos
3 1
3cos
j2 ( ) = 3 sin
, etc.
2
,
Turning now to the behavior of the jl ( ) s for large , from
l
1 d sin
jl ( ) = ( )
d
l
it is evident that the dominant term in the large regime (the one of order 1/) 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
jl ( )
l
sin as .
2
1
These jl ( ) , then, are the physical partial-wave solutions to the Schrdinger equation with zero
potential. When a potential is turned on, the wave function near the origin is still l
(assuming, as we always do, that the potential is negligible compared with the l ( l + 1) / 2 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
jl ( ) above except that there will be an extra phase factor, called the phase shift and denoted
by ) 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
Schrdinger 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
n0 ( ) =
the minus sign being the standard convention.
cos
,
7
An argument parallel to the one above for the Bessel functions establishes that the higher-order
Neumann functions are given by:
l
1 d cos
nl ( ) = ( )
.
d
l
Near the origin
nl ( )
( 2l 1)!! as 0
l +1
and for large
nl ( )
l
cos as ,
2
1
l
sin + asymptotically can be written as a linear
2
combination of Bessel and Neumann functions in that region.
so a function of the form
1
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:
hl ( ) = jl ( ) + inl ( ) , hl * ( ) = jl ( ) inl ( ) ,
so for large ,
hl ( )
ei( l / 2)
e i( l / 2)
, hl * ( )
.
i
i
The Partial Wave Scattering Matrix
Let us imagine for a moment that we could just send in a (time-independent) spherical wave,
with variation given by Pl(cos). For this l th partial wave (dropping overall normalization
constants as usual) the radial function far from the origin for zero potential is
i kr l / 2 )
+ i kr l / 2 )
1
l i e (
e(
jl ( kr ) sin kr =
.
kr
2 2k
r
r
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 :
8
l
sin kr
2
l
sin kr + l ( k ) .
2
This is what we would find on integrating the Schrdinger 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 Schrdinger equation, so to match with
i k
the result above we multiply the whole partial wave function by the phase factor e l ( ) . 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
i e
2k
i ( kr l / 2 )
r
e
+ i ( kr l / 2 )
r
i kr l / 2 )
S ( k ) e + i( kr l / 2)
i e (
l
.
2k
r
r
This equation introduces the scattering matrix
Sl ( k ) = e 2il ( k ) ,
which must lie on the unit circle |S| =1 to conserve probabilitythe 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 Schrdingers equation is solved for a nonzero potential
by integrating out from the origin, with = 0 and = 1 initially, the real function thus
generated differs from the wave function given above by an overall phase factor e
i l ( k )
.
Scattering of a Plane Wave
Were 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
eik .r = eikr cos = i l (2l + 1) jl ( kr ) Pl (cos ) = i l (2l + 1)
l
l
i e
2k
i ( kr l / 2 )
Switching on the potential phase shifts factor the outgoing wave:
r
e
+ i ( kr l / 2 )
r
Pl (cos ).
9
e
Sl ( k ) e (
r
+ i kr l / 2 )
+ i ( kr l / 2 )
r
The actual scattering by the potential is the difference these between two terms. The complete
wave function in the far region (including the incoming plane wave) is therefore:
( r , , ) = eikr cos + ( 2l + 1)
l
( S ( k ) 1) P
eikr
l ( cos )
l
2ik
r
.
The i l factor cancelled the e il / 2 . The -1 in ( Sl ( k ) 1) is there because zero scattering means
S = 1 . Alternatively, it could be regarded as subtracting off the outgoing waves already present
in the plane wave, as discussed above. There is no -dependence since with the potential being
spherically-symmetric the whole problem is azimuthally-symmetric about the direction of the
incoming wave.
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 wavea
tautology, perhaps. But if an extra phase is introduced into the outgoing wave only, the ingoing
+ outgoing will no longer give a plane wavethere will be an extra outgoing part proportional to
( Sl ( k ) 1) .
Recall that the scattering amplitude f ( , ) was defined in terms of the solution to
Schrdingers equation having an ingoing plane wave by
( r , , ) = e
ikr cos
eikr
+ f ( , )
.
r
Were now ready to express the scattering amplitude in terms of the partial wave phase shifts (for
a spherically symmetric potential, of course):
f ( , ) = f ( ) = ( 2l + 1)
l
( S ( k ) 1) P
l
2ik
l
( cos ) = ( 2l + 1) fl ( k ) Pl ( cos )
l
where
fl ( k ) =
1 i l ( k )
sin l ( k )
e
k
is called the partial wave scattering amplitude, or just the partial wave amplitude.
10
So the total scattering amplitude is the sum of these partial wave amplitudes:
f ( ) =
1
( 2l + 1) eil ( k ) sin l ( k ) Pl ( cos ).
kl
The total scattering cross-section
= f ( ) 2 d
= 2
f ( )
2
sin d
0
= 2
0
2
1
( 2l + 1) eil ( k ) sin l ( k ) Pl ( cos ) sin d
kl
and the normalization of the Legendre polynomials
2
P ( cos ) sin d = 2l + 1
2
l
0
gives
= 4 ( 2l + 1) f l ( k ) =
l =0
2
4
k2
( 2l + 1) sin
l =0
2
l .
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 valuesthe 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 ( 4 / k 2 ) ( 2l + 1) , 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 L = l and p = k so l = rk. Therefore the annular area
corresponding to angular momentum between l and l + 1 has inner and outer radii l / k and
( l + 1) / k and therefore area ( 2l + 1) / k 2 .) The quantum result is essentially a diffractive effect,
well discuss it more later.
Its easy to prove the optical theorem for a spherically-symmetric potential: just take the
imaginary part of each side of the equation
f ( ) =
1
( 2l + 1) eil ( k ) sin l ( k ) Pl ( cos )
kl
11
at = 0, using Pl (1) = 1,
Im f ( = 0 ) =
1
( 2l + 1) sin 2 l ( k )
kl
from which the optical theorem Im f ( 0 ) = k / 4 follows immediately.
Its also worth noting what the unitarity of the lth partial wave scattering matrix Sl Sl = 1 implies
1
for the partial wave amplitude fl ( k ) = ei l ( k ) sin l ( k ) . Since Sl ( k ) = e 2il ( k ) , it follows that
k
Sl ( k ) = 1 + 2ikfl ( k ) .
From this, Sl Sl = 1 gives:
Im f l ( k ) = k fl ( k ) .
2
This can be put more simply:
Im
1
= k .
fl ( k )
In fact,
fl ( k ) =
1
k ( cot l ( k ) i )
.
Phase Shifts and Potentials: Some Examples
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,
i e i( kr l / 2) e 2il ( k ) e + i( kr l / 2)
l (r ) =
2k
r
r
=
=
eil ( k )
sin ( kr + l ( k ) l / 2 )
kr
e
i l ( k )
kr
( sin ( kr l / 2 ) cos ( k ) + cos ( kr l / 2 ) sin ( k ) ) .
l
l
12
This expression is only exact in the limit r , but since the potential can be taken zero
beyond r = b, the wave function must have the form
l ( r ) = ei ( k ) ( cos l ( k ) jl ( kr ) sin l ( k ) nl ( kr ) )
l
for r > b.
(The - sign comes from the standard convention for Bessel and Neumann functionssee
earlier.)
The Hard Sphere
The simplest example of a scattering potential:
V ( r ) = for r < R,
V ( r ) = 0 for r R.
The wave function must equal zero at r = R, so from the above form of l ( r ) ,
tan l ( k ) =
For l = 0,
tan 0 ( k ) =
jl ( kR )
.
nl ( kR )
( sin kR ) / kR = tan kR,
( cos kR ) / kR
so 0 ( k ) = kR. This amounts to the wave function being effectively moved over to begin at R
instead of at the origin:
sin k ( r + ) sin k ( r R )
sin kr
=
kr
kr
kr
for r > R, of course = 0 for r < R.
For higher angular momentum states at low energies (kR << 1),
j ( kR )
( kR ) / ( 2l + 1)!! =
( kR )
tan l ( k ) = l
.
l +1
2
nl ( kR )
( 2l 1)!!/ ( kR )
( 2l + 1) ( ( 2l 1)!!)
l
2 l +1
Therefore at low enough energy, only l = 0 scattering is importantas is obvious, since an
incoming particle with momentum p = k and angular momentum l 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.
13
The Born Approximation for Partial Waves
From the definition of f ( , )
k (r ) = ei k r + f ( , )
ei k r
r
and
k ( r ) = e i k r
m ei kr
i k f r
3
2
d r e V (r ) k (r )
2
r
recall the Born approximation amounts to replacing the wave function k (r ) 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 k to be in the zdirection, so in the integrand we replace k (r ) by
eikr cos = i l (2l + 1) jl (kr ) Pl (cos ).
l
Labeling the angle between k f and r by ,
e
i k f r
= ( i ) (2l + 1) jl (kr ) Pl (cos ).
l
l
Now k f is in the direction ( , ) and r in the direction ( , ) , and is the angle between
them. For this situation, there is an addition theorem for spherical harmonics:
Pl ( cos ) =
4 l *
Ylm ( , ) Ylm ( , ) .
2l + 1 m = l
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 k (r ) expansion, giving
f ( ) =
2m
2
( 2l + 1)P ( cos ) r drV ( r ) ( j ( kr ) )
2
l =0
l
2
l
0
and remembering
f ( ) =
1
( 2l + 1) eil ( k ) sin l ( k ) Pl ( cos )
kl
it follows that for small phase shifts (the only place its valid) the partial-wave Born
approximation reads
14
l ( k )
2mk
r drV ( r ) ( j ( kr ) )
2
l
2
2
.
0
Low Energy Scattering: the Scattering Length
From
fl ( k ) =
k ( cot l ( k ) i )
the l = 0 cross section is
l =0 =
1
4
k 2 cot 0 ( k ) i
2
,
.
At energy E 0, the radial Schrdinger equation for u = r away from the potential becomes
d 2u / dr 2 = 0 , with a straight line solution u ( r ) = C ( r a ) . This must be the k 0 limit of
u ( r ) = C sin ( kr + 0 ( k ) ) , which can only be correct if 0 is itself linear in k for sufficiently
small k, and then it must be 0 ( k ) = ka, 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
l =0 ( k 0 ) = 4 a 2 .
The quantity a is called the scattering length.
Integrating the zero-energy radial Schrdinger 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 its a straight line outside of the well!):
15
Scattering Length for Square W ell
2
1
0
-5
0
5
10
-1
-2
Potential
Tangent
Wavef unction
Attractive potential:
S cattering Length for Square W ell
2
1
0
-5
0
5
10
-1
-2
Potential
Tangent
Wavef unction
On increasing the strength of the repulsive potential, still solving for the zero-energy wave
function, a tends to the potential wallheres the zero-energy wavefunction for a barrier of
height 6:
16
Scattering Length for Square W ell
1
0
-5
0
5
10
-1
Potential
Tangent
Wavef unction
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 a2.
On increasing the strength of the attractive well, if there is a phase change greater that /2 within
the well, a will become positive. In fact, right at /2, a is infinite!
S cattering Length for Square W ell
2
1
0
-5
0
5
-1
-2
Potential
Tangent
Wavef unction
And a little more depth to the well gives a positive scattering length:
10
17
Scattering Length for Square W ell
2
1
0
-5
0
5
10
-1
-2
Potential
Tangent
Wavef unction
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 outsidea genuine bound
state, with wave function e r outside the well.
0.5
0
0
5
10
15
- 0.5
Potential
Wavef unction
20
25
30
18
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 1 / a , 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,
B. E. =
2
k 2 / 2m =
2
/ 2ma 2 .
(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.