\chapter{Background}
\label{chap:Background}

\section{Partial-wave decomposition}
\label{sec:partial_wave}

For spherical symmetric two-body potentials $V(r)$, where r is the distance between the particles, the Hamiltonian has an $SO(3)$ symmetry. Consequently, the time-independent Schr\"odinger equation,
\begin{equation}
    \left[{-}\frac{1}{2\mu}\nabla^2 + V(r)\right]\psi(\vec{x}) = E \, \psi(\vec{x}) \, ,
    \label{eq:SE}
\end{equation}
can be separated into radial and angular parts by writing out the Laplacian operator in spherical coordinates,
\begin{equation}
    \nabla^2 \equiv \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) + \frac{1}{r^2} \left[ \frac{1}{\sin \theta}  \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right] \, ,
\end{equation}
which gives the angular eigen-equation,
\begin{equation}
    {-}\left[ \frac{1}{\sin \theta}  \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right] Y^m_l(\theta,\phi) = l(l+1) \, Y^m_l(\theta,\phi) \, ,
\end{equation}
having the well-known spherical harmonics $Y^m_l$ as eigenfunctions and $l(l+1)$ as corresponding eigenvalues. Substituting this back into Eq.~\eqref{eq:SE} gives the radial Schr\"odinger equation,
\begin{equation}
    \left[ \frac{\dd ^2}{\dd r^2} - \frac{l(l+1)}{r^2} + 2 \mu V(r) - p^2 \right] u(r) = 0 \, ,
    \label{eq:SE_rad}
\end{equation}
where $p=\sqrt{2 \mu E}$ and $u(r)$ is the \emph{reduced} radial wave function implicitly defined via,
\begin{equation}
    \psi(r,\theta,\phi) = R(r) Y^m_l(\theta,\phi) = \frac{u(r)}{r} Y^m_l(\theta,\phi) \, .
\end{equation}

\subsection{Free solutions}

In a region of zero-interaction, that is $V(r)=0$, Eq.~\eqref{eq:SE_rad} has solutions that are given by some linear combination of Ricatti-Hankel functions,
\begin{equation}
 u(r) \sim a \, \hat{h}^-_l(pr) + b \, \hat{h}^+_l(pr) \, ,
 \label{eq:free_wf}
\end{equation}
where $\hat{h}^+_l(pr)$ and $\hat{h}^-_l(pr)$ are the Ricatti-Hankel functions of the first and second kind respectively. These Ricatti-Hankel functions can be defined using the familiar spherical Bessel functions,
\begin{subalign}
    \hat{h}^+_l(z) &= \ii z \, h^{(1)}_l(z) \\
    \hat{h}^-_l(z) &= {-}\ii z \, h^{(2)}_l(z)
\end{subalign}

When $V(r)=0$ everywhere, we can fix the coefficients $a$ and $b$ to satify the boundary condition $u(r) \to 0$ for $r \to 0$. This gives the ``free solutions'' of the the radial Schr\"odinger equation,
\begin{equation}
 u(r) = \hat{j}_l(pr) = \frac{i}{2} \left[\hat{h}^-_l(pr) - \hat{h}^+_l(pr)\right] \, .
\end{equation}
$\hat{j}_l(pr)$ are called the Ricatti-Bessel functions and they form a complete basis for any given $l$.

\subsection{The radial basis}

Next, we define the partial-wave ``radius'' state $\ket{r,l,m}$ such that,
\begin{equation}
    \braket{r,l,m | \psi_{l,m}} = u(r) \, ,
\end{equation}
where $\psi_{l,m}$ is a state with definite angular momentum quantum numbers $(l,m)$ and $u(r)$ is the corresponding reduced radial wave function.
This leads to the relation,
\begin{equation}
    \braket{\vec{x} | r,l,m} = \frac{\delta (x - r)}{r} Y_l^m(\hat{x}) \, .
\end{equation}

Now, consider a free solution $\ket{E,l,m}$ with energy $E=p^2/\mu$. Its overlap with the radius state $\ket{r,l',m'}$ is,
\begin{spliteq}
    \braket{r,l',m' | E,l,m} &= \int d^3 x \, \braket{r,l',m' | \vec{x}} \braket{\vec{x} | E,l,m} \\
    &= \iiint x^2 \, dx \, d(\cos{\theta}) \, d\phi \, \frac{\delta (x - r)}{r} Y_{l'}^{m'}(\theta,\phi) \left[\ii^l \sqrt{\frac{2\mu}{\pi p}} \frac{1}{r} \hat{j}_l(pr) Y_l^m(\theta,\phi) \right] \\
    &= \ii^l \sqrt{\frac{2\mu}{\pi p}} \hat{j}_l(pr) \, \delta_{l,l'} \, \delta_{m,m'}
\end{spliteq}
where the relation
\begin{equation}
    \braket{\vec{x} | E,l,m} = \ii^l \sqrt{\frac{2\mu}{\pi p}} \frac{1}{r} \hat{j}_l(pr) Y_l^m(\theta,\phi) 
\end{equation}
is used~\citep[Eq. 11.8]{taylor}.

\section{Hankel transform}

We can transform between the ``partial-wave momentum space'' and the usual partial-wave configuration space using the Hankel transforms. We adopt the ``symmetric'' convention given by,
\begin{subalign}
    u(k) &= \sqrt{\frac{2}{\pi}} \int_0^\infty dr \, \hat{j}_l(kr) u(r) \, ,\\
    u(r) &= \sqrt{\frac{2}{\pi}} \int_0^\infty dk \, \hat{j}_l(kr) u(k) \, .   \label{eq:inv_hankel}
\end{subalign}

To verify the constant pre-factors, we show that the normalization is preserved under this transform. We know that in configuration space, the radial part of the wave function obeys the normalization condition,

\begin{equation} \begin{split}
    1 &= \int_0^\infty \, dr \, r^2 \, R^*(r) \, R(r) \\
      &= \int_0^\infty \, dr \, u^*(r) \, u(r)
\end{split} \end{equation}

And in momentum space, it obeys,

\begin{equation}
    1 = \int_0^\infty dk \, u^*(k) \, u(k)
\end{equation}

Now, we show that we end up with a properly normalized $u(r)$ in configuration space,
given we start with a normalized $\Tilde{u}(k)$ in momentum space.

\begin{equation} \begin{split}
    \int_0^\infty \, dr \, u^*(r) u(r)
    &= \frac{2}{\pi} \int_0^\infty \int_0^\infty \int_0^\infty
        dr \, dk \, dk' \,
        \hat{j}_l(kr) \, \hat{j}_l(k'r) \, u^*(k) \, u(k') \\
    &= \int_0^\infty \int_0^\infty
        dk \, dk' \,
        \delta(k-k') \, u^*(k) \, u(k') \\
    &= \int_0^\infty
        dk \, u^*(k) \, u(k) \\
    &= 1 \\
\end{split} \end{equation}

where we have used Eq.~\eqref{eq:inv_hankel} in  the first step
and evaluated the integral w.r.t. $r$ using \cite[p183]{taylor}
$$\int_0^\infty dr \, \hat{j}_l(kr) \, \hat{j}_l(k'r) = \frac{\pi}{2} \delta(k-k')$$
in the second step.