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\centerline{\bf Scattering amplitude and partial waves}
\bigskip
In assignment 4 you are asked to simulate a scattering experiment. Your
simulation will need to have some random error in the data. Before we
discuss how to produce a Gaussian probability distribution for the
random errors, let's go over the physics of scattering.
The differential cross section, $d\sigma /d \Omega$, can be written
in terms of a scattering amplitude, $f(\theta , \phi )$. If the interaction
is spherically symmetric, then there is no $\phi$ dependence for $f$. So
\begin{equation}
{{d\sigma } \over {d \Omega}} = |f(\theta )|^2
\end{equation}
\noindent where $f(\theta )$ is a complex number with units of length. In
our application, $f(\theta )$ will have units of $fm^2$. For non-relativistic
energies, $f(\theta )$ can be determined from the Schroedinger equation with
the appropriate scattering boundary conditions at $r = \infty$. If the
interaction is spherically symmetric, i.e. $V(\vec{r} = V(r)$, then the
Schroedinger equation can be separated into the different orbital
angular momentum quantum numbers $l$. We derived this separation for
the first assignment when we treated bound states. We obtained the
set of equations:
\begin{equation}
-{{\hbar^2}\over{2m}} ( {{d^2 u(r)} \over {d r^2}}
- {{l(l+1)} \over {r^2}} u(r) )
+ V(r) u(r) = E u(r)
\end{equation}
\noindent with an equation for each value of $l$. The same separation holds
for scattering problems. One will obtain a scattering amplitude for each
value of orbital angular momentum $l$, which we label as $f_l$. The
complete scattering solution will have a $Y_{lm}(\theta, \phi)$ added on
for each $l$. For spherical symmetry, where there is no $\phi$ dependence, so
the $Y_{lm}$ reduce to Legendre polynomicals in $cos(\theta)$, $P_l(\theta )$.
The scattering amplitude therefore becomes
\begin{equation}
f(\theta ) = \sum_{l=0}^\infty (2l+1)f_l P_l(cos(\theta ))
\end{equation}
\noindent The scattering amplitude $f(\theta )$ and the $f_l$ are
complex numbers, with a real and an imaginary part. \\
The sum over orbital angular momentum $l$ in the expression for the scattering amplitude
goes to infinity. The contribution from large $l$ goes to zero, and one only
needs to sum over a few values of $l$. The maximum value needed for $l$ is
roughly $Rpc$, where $R$ is the size of the target and $p$ is the momentum of the
projectile. In our application we will only sum over the $l=0$ and $l=1$ {\bf "partial
waves"}.\\
One can express the partial wave amplitudes $f_l$ in terms of "phase shifts". The
phase shifts are determined from the solution of the Schroedinger equation for each
$l$ value. In terms of the {\bf phase shifts}, $\delta_l$, the elastic scattering amplitudes are
\begin{equation}
f_l = {{e^{i\delta_l}sin(\delta_l )} \over k}
\end{equation}
\noindent where $k=p/\hbar$.
You might wonder where the name phase shift comes from. The $\delta_l$ are
the shift in phase from the free particle solutions of the Schroedinger equation.
In the absence of the potential $V(r)$ ($V(r)=0$), the solutions to the
Schroedinger equation for orbital angular momentum $l$ are the spherical
Bessel functions, $j_l(kr)$. In fact, a plane wave traveling in the z-direction
expressed in spherical coordinates is given by:
\begin{equation}
e^{ikz} = \sum_{l=0}^\infty (2l+1)i^lj_l(kr) P_l(cos(\theta ))
\end{equation}
\noindent where $\theta$ is the angle with respect to the z-axis.
For large $r$, the spherical Bessel function $j_l(kr) \rightarrow sin(kr-l\pi /2)/(kr)$.
The effect of a real potential $V(r)$ is to cause a phase shift in this large $r$
limit: $sin(kr-l\pi /2 + \delta_l)/(kr)$. To solve for the phase shifts $\delta_l$,
one just iterates the Schroedinger equation to large $r$, like we did for the bound
state assignment. However for the scattering calculation, the energy is greater than
zero, and the solution oscillates. One can obtain the phase shifts by examining the
large $r$ solution to the discrete Schroedinger equation. We will not solve
the Schroedinger equation for the $\delta_l$ in our assignment 4. I'll give you
$\delta_0$ and $\delta_1$ for a particular energy, and you will generate simulated data.\\
\newpage
\begin{figure}
\includegraphics[width=14cm]{phases.png}
\end{figure}
\newpage
\begin{figure}
\includegraphics[width=14cm]{pipro.png}
\end{figure}
In the last lecture we came up with the formula for the scattering amplitude
in terms of the phase shifts. The result for {\bf elastic scattering} is
\begin{equation}
f(\theta ) = \sum_{l=0}^\infty (2l+1)f_l P_l(cos(\theta ))
\end{equation}
\noindent where
\begin{equation}
f_l = {{e^{i\delta_l}sin(\delta_l )} \over k}
\end{equation}
\noindent where $\delta_l$ are the phase shifts, and $k=p/\hbar$. For the
case of elastic scattering the $\delta_l$ are real. In our
assignment, we only sum up to $l=1$ since the momentum of the pion is
small. In this case, the formula for $f(\theta )$ is
\begin{equation}
f(\theta ) = {1 \over k} (e^{i\delta_0}sin(\delta_0)+3e^{i\delta_1}sin(\delta_1)cos(\theta) )
\end{equation}
\bigskip
Before we go over coding with complex numbers let me discuss where the name
phase shift comes from. The $\delta_l$ are
the shift in phase from the free particle solutions of the Schroedinger equation.
In the absence of the potential $V(r)$ ($V(r)=0$), the solutions to the
Schroedinger equation for orbital angular momentum $l$ are the spherical
Bessel functions, $j_l(kr)$. In fact, a plane wave traveling in the z-direction
expressed in spherical coordinates is given by:
\begin{equation}
e^{ikz} = \sum_{l=0}^\infty (2l+1)i^lj_l(kr) P_l(cos(\theta ))
\end{equation}
\noindent where $\theta$ is the angle with respect to the z-axis.
For large $r$, the spherical Bessel function $j_l(kr) \rightarrow sin(kr-l\pi /2)/(kr)$.
The effect of a real potential $V(r)$ is to cause a phase shift in this large $r$
limit: $sin(kr-l\pi /2 + \delta_l)/(kr)$. To solve for the phase shifts $\delta_l$,
one just iterates the Schroedinger equation to large $r$, like we did for the bound
state assignment. However for the scattering calculation, the energy is greater than
zero, and the solution oscillates. One can obtain the phase shifts by examining the
large $r$ solution to the discrete Schroedinger equation. We will not solve
the Schroedinger equation for the $\delta_l$ in our assignment 4. I'll give you
$\delta_0$ and $\delta_1$ for a particular energy, and you will generate simulated data.\\
The sign of the phase shift for elastic scattering at low momentum depends on whether
the interaction is attractive or repulsive. For an attractive interaction, the
"wave function" curves more than in the absence of an interaction. The result is
that the phase of the "wave function" ends up ahead of the "free particle" case.
Thus, a {\bf positive phase shift} means that the interaction (for the particular value of
$l$) is {\bf attractive}. For a repulsive interaction, the "wave function" curves less
than the free particle case, and the phase shift lags. A {\bf negative phase shift}
indicates that the interaction is repulsive. I'll demonstrate this on the board
for the $l=0$ partial wave.\\
The amplitude is complex, so in your code you will need to use complex numbers
where needed. In gcc, one need to include $<$complex.h$>$:\\
\noindent $\#$include $<$complex.h$>$\\
$\#$include $<$math.h$>$\\
\bigskip
\noindent You will need to declare any complex variables as complex:\\
\noindent complex f;\\
\bigskip
\noindent Some commands that you might need are cexp(), which is the
complex exponential function, and cabs(), which is the complex absolute value squared.
For example:
\begin{eqnarray*}
cexp(x) & \rightarrow & e^{x} \\
cabs(f) & \rightarrow & \sqrt{f^*f}
\end{eqnarray*}
Also, you will need to find the center of mass momentum, since $k=p_{cm}/\hbar$.
Now you are ready to write your code for assignment 4. I'd recommend
to first produce the data without any "Gaussian" scatter. Then throw
the Gaussian dice to simulate the statistical error in a real experiment for
each data point.
\end{document}