Quantum Mechanics · Chapter 12

The hydrogen atom

The problem that made quantum mechanics believable, solved exactly. Atomic units, the radial equation and the series that must stop, the spectrum with its lines, the degeneracy that is bigger than rotations explain, the radial wavefunctions and the orbitals built from them, expectation values, and the ions and isotopes that scale the same answer.

V (r) = - \frac{e ^{2}}{4 π ε _{0} r}

E_{n} = - \frac{13.6 eV}{n ^{2}}

a_{0} = \frac{4 π ε _{0} ℏ ^{2}}{m _{e} e ^{2}}

ψ_{n ℓ m} = R_{n ℓ} Y_{ℓ m}

Lessons

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05Radial wavefunctionsThe radial functions of hydrogen are polynomials times decaying exponentials. The most telling picture is not $R$ itself but the radial probability density $r^{2}R^{2}$, the chance of finding the electron at distance $r$ in any direction. For 1s it peaks at exactly the Bohr radius; 2s has an inner bump and a node before its main peak; 2p, with no node, peaks at four Bohr radii. The higher $n$, the further out the electron lives, roughly as $n^{2}$.Read lesson →

06OrbitalsA hydrogen state is a radial function times a spherical harmonic, and its probability density is a cloud in space. s orbitals are spherical; p orbitals are two lobes on either side of a nodal plane; d orbitals have four lobes or two lobes and a ring. Chemists draw the real combinations — $p_{x}$, $p_{y}$, $p_{z}$ — because they point along axes and form bonds, while physicists often use the states of definite $m$. Each orbital has $n-1$ nodal surfaces in all: $n-\ell-1$ spheres from the radial part and $\ell$ from the angular part.Read lesson →

07Expectation values in hydrogenAverages tell what a measurement on many atoms would find. The mean distance of a 1s electron is one and a half Bohr radii, more than the most probable distance because the tail reaches far. The mean potential energy is exactly twice the total energy, and the mean kinetic energy exactly minus it: the virial theorem of the Coulomb force, the same balance as for planets. So a hydrogen atom in its ground state has 13.6 eV of kinetic energy and $-27.2$ eV of potential energy.Read lesson →

08Hydrogen-like ions and isotopesStrip an atom of all but one electron and it is hydrogen again, with a nucleus of charge $Ze$. Nothing needs solving afresh: replace $e^{2}$ by $Ze^{2}$ and every result scales. The energies grow as $Z^{2}$, the sizes shrink as $1/Z$. Helium with one electron is bound four times as strongly as hydrogen, uranium with one electron some eight thousand times. Changing the orbiting particle instead — a muon, or a positron’s partner — changes the reduced mass and scales everything again.Read lesson →

09Chapter practiceBring the chapter together: atomic units, the termination of the radial series, the spectrum and its series, the $n^{2}$ degeneracy and its hidden symmetry, radial functions and densities, orbitals and their nodes, averages and the virial theorem, and one-electron ions. No worked example sits above the answers.Read lesson →

Mapping the lesson.

The hydrogen atom

Build the foundation