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Consider time-independent, energy conserving scattering in which the Hamiltonian
of the system is written
![\begin{displaymath}
H = H_0 + V({\bf r}),
\end{displaymath}](img2835.png) |
(1250) |
where
![\begin{displaymath}
H_0 = \frac{p^2}{2 m} \equiv - \frac{\hbar^2}{2 m} \nabla^2
\end{displaymath}](img2836.png) |
(1251) |
is the Hamiltonian of a free particle of mass
, and
the scattering potential. This potential is assumed to only be
non-zero in a fairly localized region close to the origin. Let
![\begin{displaymath}
\psi_0({\bf r}) = \sqrt{n} {\rm e}^{ {\rm i} {\bf k}\cdot {\bf r}}
\end{displaymath}](img2838.png) |
(1252) |
represent an incident beam of particles, of number density
, and
velocity
. Of course,
![\begin{displaymath}
H_0 \psi_0= E \psi_0,
\end{displaymath}](img2840.png) |
(1253) |
where
is the particle energy.
Schrödinger's equation for the scattering problem is
![\begin{displaymath}
(H_0+V) \psi = E \psi,
\end{displaymath}](img2842.png) |
(1254) |
subject to the boundary condition
as
.
The above equation can be rearranged to give
![\begin{displaymath}
(\nabla^2+k^2) \psi = \frac{2 m}{\hbar^2} V \psi.
\end{displaymath}](img2845.png) |
(1255) |
Now,
![\begin{displaymath}
(\nabla^2+k^2) u({\bf r}) = \rho({\bf r})
\end{displaymath}](img2846.png) |
(1256) |
is known as the Helmholtz equation. The solution to this
equation is well-known:
![\begin{displaymath}
u({\bf r}) = u_0({\bf r}) - \int \frac{{\rm e}^{ {\rm i} k...
...\pi \vert{\bf r}-{\bf r}'\vert} \rho({\bf r}') d^3{\bf r}'.
\end{displaymath}](img2847.png) |
(1257) |
Here,
is any solution of
.
Hence, Eq. (1255) can be inverted, subject to the boundary condition
as
, to give
![\begin{displaymath}
\psi({\bf r}) = \psi_0({\bf r})- \frac{2 m}{\hbar^2}
\int\f...
...f r}-{\bf r}'\vert} V({\bf r}') \psi({\bf r}') d^3{\bf r}'.
\end{displaymath}](img2850.png) |
(1258) |
Let us calculate the value of the wavefunction
well outside the
scattering region. Now, if
then
![\begin{displaymath}
\vert{\bf r}-{\bf r}'\vert \simeq r - \hat{\bf r}\cdot {\bf r}'
\end{displaymath}](img2853.png) |
(1259) |
to first-order in
, where
is a unit vector
which points from the scattering region to the observation point.
It is helpful to define
. This is the wavevector
for particles with the same energy as the incoming particles (i.e.,
) which propagate from the scattering region to the observation
point. Equation (1258) reduces to
![\begin{displaymath}
\psi({\bf r}) \simeq \sqrt{n}\left[{\rm e}^{ {\rm i} {\bf ...
...
+ \frac{e^{ {\rm i} k r}}{r} f({\bf k}, {\bf k}')\right],
\end{displaymath}](img2858.png) |
(1260) |
where
![\begin{displaymath}
f({\bf k},{\bf k}') = -\frac{m}{2\pi \sqrt{n} \hbar^2}\int...
...f k}'\cdot{\bf r}'} V({\bf r}') \psi({\bf r}') d^3{\bf r}'.
\end{displaymath}](img2859.png) |
(1261) |
The first term on the right-hand side of Eq. (1260) represents the incident particle
beam, whereas the second term represents an outgoing spherical wave
of scattered particles.
The differential scattering cross-section
is
defined as the number of particles per unit time scattered into
an element of solid angle
, divided by the incident
particle flux. From Sect. 7.2, the probability flux (i.e., the
particle flux) associated with a wavefunction
is
![\begin{displaymath}
{\bf j} = \frac{\hbar}{m} {\rm Im}(\psi^\ast \nabla\psi).
\end{displaymath}](img2862.png) |
(1262) |
Thus, the particle flux associated with the incident wavefunction
is
![\begin{displaymath}
{\bf j} = n {\bf v},
\end{displaymath}](img2863.png) |
(1263) |
where
is the velocity of the incident
particles. Likewise, the particle flux associated with the scattered
wavefunction
is
![\begin{displaymath}
{\bf j}' = n \frac{\vert f({\bf k},{\bf k}')\vert^2}{r^2} {\bf v}',
\end{displaymath}](img2865.png) |
(1264) |
where
is the velocity of the scattered particles.
Now,
![\begin{displaymath}
\frac{d\sigma}{d\Omega} d\Omega = \frac{r^2 d\Omega \vert{\bf j}'\vert}{\vert{\bf j}\vert},
\end{displaymath}](img2867.png) |
(1265) |
which yields
![\begin{displaymath}
\frac{d\sigma}{d\Omega} = \vert f({\bf k},{\bf k}')\vert^2.
\end{displaymath}](img2868.png) |
(1266) |
Thus,
gives the differential cross-section
for particles with incident velocity
to be scattered such that their final velocities are directed into a range of
solid angles
about
. Note that the scattering
conserves energy, so that
and
.
Next: Born Approximation
Up: Scattering Theory
Previous: Introduction
Richard Fitzpatrick
2010-07-20