Hydrogen is the fundamental three-dimensional bound-state problem in quantum mechanics. After separating center-of-mass motion, the electron-proton relative coordinate has reduced mass $\mu$ and Coulomb potential $$V(r)=-\frac{e^2}{4\pi\varepsilon_0r}.$$ Because the potential is spherically symmetric, stationary states separate into radial and angular parts: $$\psi_{n\ell m}(r,\theta,\phi)=R_{n\ell}(r)Y_{\ell m}(\theta,\phi).$$ The angular functions are spherical harmonics and carry orbital angular-momentum quantum numbers $\ell$ and $m$.

Solving the radial equation with normalizable boundary conditions gives discrete energies $$E_n=-\frac{\mu e^4}{2(4\pi\varepsilon_0)^2\hbar^2}\frac{1}{n^2}, \qquad n=1,2,3,\ldots$$ Numerically, using the electron mass approximation, $$E_n\approx-\frac{13.6,\mathrm{eV}}{n^2}.$$ In the nonrelativistic Coulomb problem the energy depends only on $n$, producing degeneracy among different $\ell$ and $m$ values with the same principal quantum number.

The ground state has $n=1$, $\ell=0$, $m=0$ and a characteristic length equal to the Bohr radius $$a_0=\frac{4\pi\varepsilon_0\hbar^2}{\mu e^2}.$$ Its probability density is largest near the nucleus but remains spatially extended; the electron is not moving on a classical planetary orbit.

Hydrogen connects quantum mechanics to atomic spectra, angular momentum and chemistry. More detailed structure—spin, relativistic effects and external fields—splits the idealized degeneracies but builds on this same central Coulomb problem.