We have the following wavefunction for the hydrogen atom:
$$\psi(r,\theta,\phi)=\frac{1}{\sqrt{4\pi}}\frac{1}{(2a)^{3/2}}\frac{r}{a}e^{-r/2a}\sin(\theta)\sin(\phi)$$
where $a$ is the Bohr radius.
Question
: How can I express the wavefunction above as a linear combination of the eigenstates of the hydrogen atom's Hamiltonian? In other words, we need to express the above in terms of a linear combination of
$$\psi_{nlm}=\sqrt{\left(\frac{2}{na}\right)^3 \frac{(n-l-1)!}{2n[(n+l)!]^3}}e^{-r/na}\left(\frac{2r}{na}\right)[L_{n-l-1}^{2l+1}(2r/na)]Y_l^m(\theta,\phi)$$
For some $n$, $l$, and $m$'s. How can I find such a linear combination without just a simple guess and check; i.e., what is a systematic way to find the linear combination?