Hydrogen

The Hydrogen (Coulomb potential) radial equation is solved by finding the behavior at large $r$, then finding the behavior at small $r$, then using a power series solution to get

$$\bgroup\color{black}R(\rho)=\rho^\ell\sum\limits_{k=0}^\infty a_k\rho^k e^{-\rho/2}\egroup$$

with $\rho=\sqrt{-8\mu E\over\hbar^2}r$. To keep the wavefunction normalizable the power series must terminate, giving us our energy eigenvalue condition.

$$\bgroup\color{black}E_n=-{Z^2\alpha^2mc^2\over 2n^2}\egroup$$

Here $n$ is called the principle quantum number and it is given by

$$\bgroup\color{black}n=n_r+\ell+1\egroup$$

where $n_r$ is the number of nodes in the radial wavefunction. It is an odd feature of Hydrogen that a radial excitation and an angular excitation have the same energy.

So a Hydrogen energy eigenstate $\psi_{n\ell m}(\vec{x})=R_{n\ell}(r)Y_{\ell m}(\theta,\phi)$ is described by three integer quantum numbers with the requirements that $n\geq 1$, $\ell<n$ and also an integer, and $-l\leq m\leq\ell$. The ground state of Hydrogen is $\psi_{100}$ and has energy of -13.6 eV. We compute several of the lowest energy eigenstates.

The diagram below shows the lowest energy bound states of Hydrogen and their typical decays.