The Coulomb Potential

The hydrogen atom is the single most important exactly solvable system in quantum mechanics. A single electron bound to a single proton is simple enough to solve in closed form, yet rich enough to predict spectral lines, orbital shapes, and the structure of the periodic table. The whole story begins with one ingredient: the electrostatic attraction between the two charges.

The classical interaction

A proton carries charge $+e$ and an electron carries charge $-e$ . Coulomb's law gives the potential energy of the pair as a function of their separation $r$ :

V(r) = -\frac{1}{4\pi\epsilon_0}\frac{e^2}{r}.

Two features matter. First, $V$ is negative: opposite charges attract, so bringing them together lowers the energy. Second, $V$ depends only on the distance $r$ , not on direction. A potential of this form is called a central potential, and that spherical symmetry is what makes the problem tractable.

The two-body to one-body reduction

A proton and electron both move, so strictly we have a two-body problem. As in classical mechanics, we separate the centre-of-mass motion from the relative motion. The internal dynamics is then that of a single fictitious particle of reduced mass

\mu = \frac{m_e\, m_p}{m_e + m_p},

moving in the potential $V(r)$ , where $r$ is now the electron–proton separation. Because the proton is about $1836$ times heavier than the electron, $\mu \approx m_e$ to better than one part in a thousand. We will usually write $m_e$ and treat the proton as a fixed centre, remembering that the exact treatment replaces $m_e$ by $\mu$ .

The hydrogen Hamiltonian

The Hamiltonian is kinetic energy plus potential energy. For the relative coordinate,

\hat{H} = -\frac{\hbar^2}{2\mu}\nabla^2 - \frac{1}{4\pi\epsilon_0}\frac{e^2}{r}.

The time-independent Schrödinger equation we must solve is

\left[-\frac{\hbar^2}{2\mu}\nabla^2 - \frac{1}{4\pi\epsilon_0}\frac{e^2}{r}\right]\psi(\mathbf{r}) = E\,\psi(\mathbf{r}).

The Laplacian $\nabla^2$ encodes the kinetic energy; the $-1/r$ term is the Coulomb attraction. Bound states have $E < 0$ (the electron is trapped in the well), while $E > 0$ describes a free electron scattering off the proton.

Why spherical coordinates

Because $V$ depends only on $r$ , the natural coordinates are spherical: $(r, \theta, \phi)$ . In these coordinates the Laplacian splits cleanly into a radial piece and an angular piece, and the angular piece is exactly the operator whose eigenfunctions are the spherical harmonics. That separation — the subject of the next lesson — is what turns a three-dimensional partial differential equation into a one-dimensional ordinary differential equation we can actually solve.

The takeaway

The hydrogen atom is governed by a single central potential $V(r) = -e^2/(4\pi\epsilon_0 r)$ , and its Hamiltonian is the corresponding kinetic-plus-potential operator written with the reduced mass $\mu$ . The spherical symmetry of the Coulomb interaction is the structural fact that makes the atom solvable, and everything in this module builds on it.