Scattering Theory
Michael Fowler 1/16/08
References: Baym, Lectures on Quantum Mechanics, Chapter 9.
Sakurai, Modern Quantum Mechanics, Chapter 7.
Shankar, Principles of Quantum Mechanics, Chapter 19.
Introduction
Almost everything we know about nuclei and elementary
particles has been discovered in scattering experiments, from
The simplest model of a scattering experiment is given by solving Schrödinger’s equation for a plane wave impinging on a localized potential. A potential V(r) might represent what a fast electron encounters on striking an atom, or an alpha particle a nucleus. Obviously, representing any such system by a potential is only a beginning, but in certain energy ranges it is quite reasonable, and we have to start somewhere!
The basic scenario is to shoot in a stream of particles, all at the same energy, and detect how many are deflected into a battery of detectors which measure angles of deflection. We assume all the ingoing particles are represented by wavepackets of the same shape and size, so we should solve Schrödinger’s time-dependent equation for such a wave packet and find the probability amplitudes for outgoing waves in different directions at some later time after scattering has taken place. But we adopt a simpler approach: we assume the wavepacket has a well-defined energy (and hence momentum), so it is many wavelengths long. This means that during the scattering process it looks a lot like a plane wave, and for a period of time the scattering is time independent. We assume, then, that the problem is well approximated by solving the time-independent Schrödinger equation with an ingoing plane wave. This is much easier!
All we can detect are outgoing waves far outside the region
of scattering. For an ingoing plane wave
, the wavefunction far away from the scattering region must
have the form
where θ, φ are measured with respect to the ingoing direction.
Note that the scattering
amplitude 
has the dimensions of
length.
We don’t worry about overall normalization, because what is
relevant is the fraction of the
incoming beam scattered in a particular direction, or, to be more precise, into
a small solid angle 
in
the direction θ, φ .
The ingoing particle current (with the above normalization) is 
through unit area
perpendicular to the ingoing beam, the outgoing current into the small angle is 
. It is evident that this
outgoing current corresponds to the original ingoing current flowing through a
perpendicular area of size 
, and
is called the differential cross section for scattering in the direction θ, φ.
The Time-Independent Description
We shall review the time-independent formulation of scattering theory, first as it is presented in Baym, in terms of the standard Schrödinger equation wavefunctions, then do the same thing a la Sakurai, in the more formal, but of course equivalent, language of bras and kets. The Schrödinger wavefunction approach is an easier introduction, but the formal language is more convenient for analyzing the structure of higher order terms.
Actually, Baym’s treatment isn’t quite time-independent, in that he uses an ingoing wavepacket, but it is one of great length, well approximated by a plane wave. Sakurai goes straight to the plane wave, and we do too. This case is very reminiscent of one-dimensional scattering, in which a plane wave from the left generates outgoing waves in both directions, and the amplitudes can be calculated from the Schrödinger equation for a single energy eigenstate. The only difference is that in 3D there will be outgoing waves in all directions.
Following Baym, Schrödinger’s equation is:
This 
we take to have an
incoming plane wave component 
. Overall
normalization is irrelevant, since the differential cross-section depends only
on the ratio of the scattered wave amplitude to that of the ingoing
wave.
The standard approach to an equation like the one above is
to transform it into an integral equation using Green’s functions. If 
is small (just how small
it has to be will become clear later) the integral equation can then be solved
by iteration.
The Green’s function 
is essentially the
inverse of the differential operator,
This is not a mathematically unique definition: clearly, we
can add to any solution of the
homogeneous equation
for example, the incoming plane wave.
If we write the integral equation
this 
is certainly a
solution to the original Schrödinger equation, as is easily checked by applying
the operator
to both sides of the equation.
The integral equation can be formally solved by iteration, and
for “small” V the solution will converge. But this won’t
really do—remember, we haven’t a unique ! We have to fix by connecting better
with the scattering problem we’re trying to solve.
We know our solution has a single ingoing plane wave, and outgoing waves in all other directions, generated by the interaction of the plane wave with the potential. But the Schrödinger equation could equally describe ingoing waves in the other directions. In defining the Green’s function and writing the integral equation, we have nowhere specified the distant form of the wavefunction, that is, we have not required that the Green’s function on the right hand side of the integral equation only generate outgoing waves. To see how to do this, we must write the Green’s function itself as a sum over waves, in other words a Fourier transform, and see how to eliminate the unphysical (for the present problem) incoming waves in that sum.
The explicit form of the Green’s function is
Note that G(r, k)
only depends on 
through Ek,
and only on 
through r,
since the integration over 
is over all
directions. It is easy to verify that this Green’s function satisfies the
differential equation, by applying the differential operator to the first
integral above: the result is to cancel the denominator in the integral,
leaving just 
, which is the d-function
in .
To get the second form of G(r, k) in the equation above, we first do the angular integration d(cosq )
to get 
, then rearrange the integral over the -e-ik′r
term by switching the sign of 
, so it becomes an integral from -∞ to 0 instead of 0
to ∞. Then we add the two terms
(the eik′r and the -e-ik′r)
together to give an integral from -∞ to ∞.
This integral from -∞ to ∞ is then done by contour
integration—at least, after we’ve figured out what to do about the
singularities at k' = ± k.
For the integral to be defined, the contour must be distorted slightly so it bypasses these poles.
It is at this point we feed in our physical knowledge of the situation: that in the scattering process, the second term in
that is, the Green’s function term, has to be a sum over outgoing waves only. And, we can guarantee this by distorting the contour of integration in the right direction, as follows.
The contour integral has to be evaluated by closing the
contour. Since r is positive eik’r
goes to zero in the upper half k'
plane, but diverges in the lower half, so we must close the contour in the
upper half plane to ensure no contribution from the semicircle at infinity.
Therefore, to get the desired outgoing waves, eikr but not
e-ikr, our contour closed in the upper half
plane must encircle the pole at k'
= +k but not the one at k' = -k. (eikr does represent
outgoing waves: the suppressed time dependence is 
, giving 
.) In other words, the
relative configuration of the real-axis part of the contour and the two poles
has to be:
x (pole)
x (pole at k' = -k - ie ) (pole at k' = +k + ie)
Instead of moving the contour slightly off the real axis to avoid the poles, we’ve moved the poles slightly instead. These movements are infinitesimal, so which gets moved makes no difference to the value of the integral. It is more convenient to move the poles, as shown, because this move can be efficiently included in the integral just by adding an infinitesimal imaginary part to the denominator:
Notice that we have written G+ instead of G, because G can denote any solution of
and we are specifying the particular solution having only outgoing waves. In contrast to G, G+ is well-defined and unique. (There is another perfectly valid solution having only ingoing waves, but it is irrelevant to the scattering problem. The difference between the ingoing and outgoing solutions satisfies the homogeneous equation having zero on the right-hand side.)
Once we move the poles slightly as described above, the pole at k' = +k + ie is in fact the only singularity of the integrand lying inside the contour of integration (closed in the upper half plane), so the value of the integral is just the contribution from this pole, that is,
.
Therefore the ie prescription (as it’s sometimes called) in G+ does indeed give us what we want: a solution having only outgoing waves, and the integral equation becomes:
This
can be written more simply if we assume the potential to be localized, so that
we can take 
. In this case, it is
a good approximation to take 
in the denominator.
However, this approximation cannot be
made in the exponential, because to leading order (see diagram)
and although the second term is much smaller than the first, it is a phase, which may be of order unity. Such a factor must of course be included so that the contributions to the integral from different regions of the potential are added with the correct relative phases.
Therefore, assuming the detector distance r is much larger than the range of the potential, we can write
The Born Approximation
From the above equation, the first order approximation to the
scattering is given by replacing y in the integral on the right with the zeroth-order
term 
,
This is the Born
approximation. In terms of the
scattering amplitude 
, which we defined in terms of the asymptotic wave function:
the Born approximation is:
where 
is the momentum transfer, 
. (Since the incoming and
outgoing momenta have equal magnitude, it is easy to check that 
)
The essential physics here is that a particle scattered with
momentum change is scattered by the 
-Fourier component of the potential—one can imagine the
potential as built up of Fourier components each of which acts like a
diffraction grating. Higher order
corrections to the Born approximation correspond to successive scatterings off
these gratings—these higher orders are generated by iteration of
It is important to establish when the Born approximation is
a good one: sometimes it isn’t. Actually, we are just doing perturbation theory
in disguise, so we need the perturbation to be small, that is to say, replacing
by 
in the integral on the
right in the equation above should only make a small difference to the value of
given by doing the
integral. This is of course a rather
tricky exercise in self-consistency.
Let us attempt to estimate what difference the replacement
of by in the integrand does make for the common case of a
spherically symmetric potential V(r) parameterized by depth V0 and range r0. The integral is effectively only over a
region of size r0 around
the origin.
First consider low
energy scattering, kr0
< 1 say, so for estimation purposes we can replace the exponential term by 1
in the region of integration. We also
assume that where y appears in the integral on
the right-hand side of the equation 
is also pretty close
to 1 (remember the integral is only over a volume within r0 or so of the origin) and so we just replace it by
1. In other words, we’re assuming that
the ingoing plane wave, the , is not dramatically distorted inside that volume where the
potential is significant.
Now, we’ve assumed the wave function near the origin is
close to 1, so putting that value in the integrand on the right had better give
a value for on the left hand side
of the equation which is pretty close to 1.
The approximations give:
so the Born approximation will be reasonable at low energies (kr0 < 1) if the second term on the right hand side is a lot less than unity.
When is this true for a real potential? Taking V(r) to have depth V0 and range r0, the Born approximation is good if:
Notice that the right hand side of this inequality is of order the kinetic energy of a particle confined to a volume equal to the range of the potential, so the Born approximation is valid at low energies provided the potential is well below the strength necessary for a bound state.
In fact, the Born approximation works better at higher
energies, because the oscillating phase term in 
cuts down the value of the integral by a factor of order of
magnitude 
. This means the condition becomes 
always satisfied at
high enough energies.
The Lippmann-Schwinger Equation
It proves illuminating, especially in understanding scattering beyond the Born approximation, to recast the Green’s function derivation of the scattering amplitude in the more formal language of bras, kets and operators. The Green’s function was introduced in the previous section as the (non-unique) inverse of the operator
.
(Parenthetical remark: in numerical computation, the wavefunction might be specified at points on a lattice in space, and a differential operator like this would be represented as a difference operator, that is, as a large but finite matrix operating on a large vector whose elements were the wavefunction values at points on the lattice. The Green’s function would then be the inverse matrix with appropriate boundary conditions specified to ensure uniqueness.)
Purely formally (and following Sakurai), writing H = H0 + V,
with H0 the kinetic energy operator 
, the ingoing plane wave state is a solution of
We want to solve
The transformation from a differential equation to an integral equation in this language is:
This gives the undisturbed incoming wave for V = 0,
and by operating on both sides of the equation with 
does indeed satisfy the full Schrödinger
equation. But of course this transformation from a differential to an integral
equation has the same flaw as the earlier treatment: H0 has a
continuum of eigenvalues in the infinite volume limit, so the operator equation
becomes ill-defined for those eigenstates with energy arbitrarily close to the
incoming energy, and those are precisely the states of physical relevance.
To make explicit that this is indeed the problem we’ve
already solved, let us translate it into the earlier language. First take the inner product with the bra 
:
.
Next, insert a representation of unity as a sum over eigenstates of momentum (and therefore of H0) into the last term:
Finally, insert another representation of unity as a sum over eigenstates of position in the last term:
Comparing
this expression with the integral equation in the earlier discussion, it is
evident that they are indeed equivalent, and therefore the correct 
prescription to give
the scattered wave function,
where
in
which form it is evident that 
is the same as 
in the previous work.
This equation for the scattered wave is called the Lippmann-Schwinger equation.
(Note: Sakurai defines his Green’s function
as 
)
Now that we have a well-defined Green’s
function operator G+, the Lippmann-Schwinger equation can be
solved formally:
with a series solution
just a formal version of the solution we found earlier.
The Transition Matrix
Operating on both sides of the above the equation with V,
defining the “transition matrix” T by
In terms of this transition matrix operator, the scattered wave can be written
Comparing this with
,
and recalling that the Born approximation is given by
,
we see that T is a kind of generalized potential, including all the higher order terms, so that just as the Born approximation gave the scattering amplitude in terms of V,
the exact result including all higher
order terms must have the same structure with T replacing V. Of course, unlike , T is not a diagonal matrix in r-space: it depends on two space variables, and its Fourier
transform is a therefore function of two momenta, that is, the incoming and the scattered . Thus we find:
We
have replaced the 
in the Born expression
with . Sakurai has an
extra (2π)3 in the term on the right, because he uses 
, 
, we use 
, 
.
The Optical Theorem
The Optical Theorem relates the imaginary part of the forward scattering amplitude to the total cross-section,
The physical content of this initially mysterious theorem will become a lot clearer after we discuss partial waves and some geometric effects. It does tell us that f cannot be real in all directions, and that in particular f has a positive imaginary part in the forward direction. We’ve included the proof here for the record, but you can skip it for now. But note that this proof is more general than the simple one given (later) in the section on partial waves, in that we do not here assume the potential to have spherical symmetry.
From the expression for above, we see that we
must find the imaginary part of 
Recall that
so we need to find
Since V is
hermitian, the only imaginary part of the above matrix element comes from the 
recalling that
Therefore,
Again using
we can rewrite the equation
Inserting a complete set of plane wave states in the final matrix element above gives
(This is the same formula as Sakurai’s in 7.3: our extra (2π)3 in the denominator is only apparent, because our plane wave states differ from his by a factor (2π)3/2. )
Time-Dependent Formulation of Scattering Theory
In the time-independent formulation presented above, we solved the Lippmann-Schwinger equation to find
where
and E = Ek.
(Reminder on our wave function normalization convention: we
always have a denominator 2π for an integral dk. This means the identity operator as a sum over plane wave
projection operators is 
. The normalization is , and . Sakurai uses , and 
as does Shankar in
Chapter 1, for one dimension, page 67, but later, Chapter 21 page 585, Shankar
has switched to our notation—so watch
out! Our convention is also used by Baym and by Peskin.)
In fact, this function