Analogy to Fourier transform in spherical coordinates with boundary at a certain radius Suppose, we have a wavefuction $\phi(\vec{x})$ which is restricted in a sphere, with the spherical boundary condtion 
$$\phi(\vec{x}=R)=\phi_0.$$
How can I do the 'Fourier transformation' as the case in a cubic box, in which case we usually can analyse the eigenvalues of momentum?
Let's be more specific. Suppose we have 
$$\phi(\vec{x})=\frac{J_1(\sqrt{R^2-|\vec{x}|^2})}{\sqrt{R^2-|\vec{x}|²},}$$
where $J_1$ is the Bessel function. This field configuaration may be totally classical. I now want to do something like the Fourier analysis, namely to figure out the components of different frequences. 
 A: The Fourier transformation is a special case of an expansion in a complete basis set. For your problem, you need the solutions of the free Schroedinger equation in spherical coordinates, which are given by
\begin{equation}
\phi_{klm}(r,\theta,\phi) = \left( a_l j_l(kr) + b_l n_l(kr) \right) Y_l^m(\theta,\phi),
\end{equation}
where $j_l$ and $n_l$ are the spherical Bessel functions, and $Y_l^m$ are the spherical harmonics.
First of all, $b_l = 0$ since $n_l$ diverges in the origin which is included in your system. The allowed values of the wave vector $k$ and the coefficients $a_l$ are fixed by the normalization and the boundary condition. Since your problem has spherical symmetry, $l$ and $m$ remain good quantum numbers. However, translation symmetry along the radial direction is broken and $k$ is no longer a good quantum number:
\begin{equation}
\phi_{nlm}(r,\theta,\phi) = a_l j_l(k_{nl} r) Y_l^m(\theta,\phi).
\end{equation}
The boundary condition $\phi(R) = \phi_0$ is not physical unless $\phi_0=0$ since the wave function is restricted inside the sphere with radius $R$ and the wave function should be  continuous.
Putting $\phi_0 = 0$, the allowed momenta are found from the zeros of the spherical Bessel function:
$$j_l(k_{nl} R) = 0,$$
where $n=1,2,\ldots$ is the radial quantum number. For $s$-waves, $l=0$ and $k_{n0} = n\pi/R$. For $l>0$, you need numerical methods to find the allowed momentum values.
