# 5.3. The optical theorem¶

We conclude this chapter by deriving an important theorem which results
from the conservation of matter. This is the *optical theorem*, which
also exists in classical optics (there it derives from total energy
conservation). To study the optical theorem, we should therefore study
the conservation of matter in quantum mechanics. For the Schrödinger
equation we can derive a material conservation law a follows. Suppose we
have a volume in three-dimensional space which is bounded by a
surface . We calculate the rate of change of the amount of
material in that space. That amount of material is given by
We then find for the rate of
change (the dot indicates derivative with respect to time):
The time-derivatives of the wave function are given by the
time-dependent Schrödinger equation:
and its hermitian conjugate
Taking for the Hamiltonian the form
, we obtain
(note that the terms containing cancel). Using Green's second
identity, the volume integral can be transformed into a surface
integral:
Here, is a vector, the norm of which is that of a small
surface segment, and directed along the outward surface normal. The
change in material can in this case only be caused by flow of material
through the boundary. In particular, we have
so that we see that the
expression for the particle flux is

Note that using the diverence theorem, the equation relating the material change to the surface integral can be related to the continuity equation: As the volume is arbitrary, this equation can only hold when

Let's now turn again to the scattering problem, where we are dealing with an incoming and an outgoing wave: The total current in this wave is then given as As this expression consists of a complex number minus its complex conjugate, we can write this in the form This can be rewritten as

In a scattering problem we look at the scattered flux generated by a stationary flux of incoming particles. In the stationary limit, there is no generation or absorption of new matter: the particles flowing into some sphere centred around the scattering potential should also come out at the same rate. Therefore we must have: as this expression calculates the total matter flux through the sphere's surface.

As we have written the total flux as a sum over an outgoing, an ingoing
and a mixing term, we can also divide the total flux through the surface
up into these three contributions. The flux of the incoming wave turns
out to be zero! This is because the beam described by
gives an incoming flux on one side of the sphere, and an *equal*
outgoing flux on the other side. The outgoing flux through the sphere is
given by
Therefore, we obtain
Using Green's theorem again, the expression on the right hand side can
be reworked:
Writing , this can be
rewritten as
Now we use that the full wave function satisfies the full Schrödinger
equation
and the incoming wave satisfies the Schrödinger equation with potential
0: Putting this
into \eqref{Eq:OptTheor} gives:
The first and the third term in the integral cancel and we have
The last term is recognised as the exact scattering amplitude for
(up to a negative pre-factor) -- see
Eq. (5.15).

Therefore, we find the *optical theorem*:
If the
wave is scattered, we see an attenuation in the forward direction
compared to the case where the incoming particles would not be
scattered. The forward scattering amplitude is therefore related
to the scattering of the particles (i.e. ). It is
important to realise that the optical theorem holds exactly; in the
first Born approximation, the theorem does not hold.