= Solution
For the nonrelativistic Coulomb <Hamiltonian> with reduced electron-proton mass $\mu$, separation in <Spherical coordinate system> gives \b[$\psi_{n\ell m}(r,\theta,\phi)=R_{n\ell}(r)Y_{\ell m}(\theta,\phi)$]. The <quantum numbers> obey $n\geq1$, $0\leq\ell\leq n-1$ and $-\ell\leq m\leq\ell$. These are joint <eigenstates> of energy, $L^2$ and $L_z$, with the latter two <eigenvalues> $\ell(\ell+1)\hbar^2$ and $m\hbar$.
The Coulomb bound-state energies and characteristic length are
$$
\boxed{E_n=-\frac{\mu e^4}{2(4\pi\epsilon_0)^2\hbar^2n^2},\qquad a_\mu=\frac{4\pi\epsilon_0\hbar^2}{\mu e^2}.}
$$
For example, $\psi_{100}=(\pi a_\mu^3)^{-1/2}e^{-r/a_\mu}$ is spherically symmetric and has no radial node. In general the radial part contains an exponential times a <Generalized Laguerre polynomial>, with $n-\ell-1$ radial nodes. The <spherical harmonic> specifies angular structure and azimuthal phase; real linear combinations of degenerate $m$ states give familiar directional orbitals.
The leading Coulomb energy depends on $n$ alone, giving $n^2$ spatial states for each principal shell, or $2n^2$ including electron <spin>. <Fine structure>, <Lamb shift> and <Hyperfine structure> lift parts of this ideal degeneracy. An <atomic orbital> is a probability amplitude, not a classical planetary path.<Quantum Mechanics by David Tong>[Section 4: separation of the Coulomb problem and hydrogenic states]{ref}
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