The Hydrogen Atom & Quantum Numbers

Chapter 5 left hydrogen with the right energies and the wrong picture: Bohr's electron circled the proton on fixed racetracks that the uncertainty principle of chapter 8 forbids. Tonight Schrödinger's equation, as chapter 10 promised, supplies the right picture. Hydrogen is the ideal patient: one proton, one electron and the electrical attraction between them. Its potential energy, V(r) = −e²/(4πε₀r), depends only on the distance r between the particles, not on direction; a potential like this is called central. The proton is 1,836 times heavier than the electron, so we treat it as fixed; letting it move shifts every energy by only about 0.05 percent.

With no preferred direction, the natural coordinates are spherical: the distance r from the nucleus, a polar angle θ measured down from a chosen z axis, and an azimuthal angle φ around that axis, like latitude and longitude on a globe. In these coordinates Schrödinger's equation separates: every solution factors into a radial part depending only on r and an angular part depending only on θ and φ. The angular parts are spherical harmonics, patterns Pierre-Simon Laplace and Adrien-Marie Legendre studied in the 1780s to describe the gravity of planets; the radial parts are polynomials times an exponential that fades with distance.

Quantization then arrives as it did for the particle in a box of chapter 11, from the demand that the wavefunction be well behaved. Going once around the z axis must bring ψ back to its starting value, so its phase can turn only a whole number of times per circuit; that whole number is mₗ. Staying finite at the poles restricts the angular pattern to a whole number l at least as large as the size of mₗ. And the radial part must fade away far from the nucleus rather than blow up, which happens only at special energies, Eₙ = −13.6 eV/n², with n a whole number greater than l. Bohr's ladder returns, derived rather than postulated.

Each solution is labelled by the whole numbers its boundary conditions produced, its quantum numbers. The principal quantum number n = 1, 2, 3 and so on fixes the energy: −13.6 eV for n = 1, then −3.40, −1.51 and −0.85 eV, crowding toward zero, where the electron breaks free. It also sets the cloud's overall size, which swells roughly as n², and its total number of nodes, n − 1, the surfaces where the wavefunction vanishes. In hydrogen the energy depends on n alone; the other numbers change the cloud's shape and orientation but not its energy.

The orbital quantum number l runs from 0 up to n − 1 and fixes the magnitude of the orbital angular momentum, the rotational momentum of a whirling object: L = √(l(l + 1))·ħ, which is 0, about 1.41ħ, 2.45ħ and 3.46ħ for l = 0, 1, 2 and 3. Here is the first nail in the coffin of the orbit. In the ground state, n = 1 forces l = 0, so the electron has no orbital angular momentum whatsoever, where Bohr had given it exactly ħ. Anything circling a centre carries angular momentum. The 1s electron carries none, so it is not circling anything.

The magnetic quantum number mₗ takes the 2l + 1 whole-number values from −l to +l and fixes the component of angular momentum along the z axis, exactly mₗħ. For any l above zero, even the largest component, lħ, is shorter than the full length √(l(l + 1))·ħ, so the angular momentum never points exactly along any axis: its components are incompatible observables, as chapter 14's commutators warned. Finally, nature adds a fourth label that Schrödinger's equation knows nothing about: spin, an intrinsic angular momentum whose measured component along any axis is always +ħ/2 or −ħ/2, recorded by the spin quantum number mₛ = ±½.

What, then, is an orbital? It is one of these solutions: a single-electron wavefunction labelled by its n, l and mₗ, a three-dimensional standing wave. By the Born rule of chapter 9, |ψ|² is its probability density, the probability per unit volume of finding the electron at a point. The chemist Robert Mulliken coined the word in 1932, short for one-electron orbital wave function, and its echo of orbit has misled students ever since. Textbook shapes are boundary surfaces enclosing, by convention, about 90 percent of the probability; the cloud itself has no edge.

Take the ground state, 1s. Its wavefunction is spherically symmetric and largest at the nucleus, shrinking by a factor of e, about 2.718, for every Bohr radius of distance; the Bohr radius, a₀, is about 0.0529 nm. So the most probable point is the nucleus itself, yet the most probable distance is a₀, because there is little room near the centre and a great deal farther out. The chance of finding the electron in a thin spherical shell of radius r, per unit of the shell's thickness, is the density times the shell's area, 4πr² times |ψ|², a quantity called the radial probability. Rising area and falling density compete, and their product peaks at exactly r = a₀: Bohr's radius survives, not as a track but as the summit of a probability distribution.

Now retire the first myth. Nothing in the 1s cloud goes round: it is a stationary state with zero angular momentum and no preferred direction to circulate in. The cloud is not a long-exposure photograph of a tiny planet but a map of the odds for where a position measurement will find the electron. Each measurement yields one point, and the cloud emerges only from many measurements on identically prepared atoms, as single dots built the double-slit fringes of chapter 7. Interpretations differ about what, if anything, the electron does between measurements, a debate for chapter 22, but all of them predict this same cloud.

Now the second. An orbital is not a container: it has no walls and holds nothing, because it is the electron's state itself. Saying an electron is in a 2p orbital describes its wavefunction, just as saying a guitar string is in its second harmonic describes how it vibrates, not a box it sits in. An empty orbital is simply an allowed state, a note the string could play.

The values of l carry letters inherited from the spectroscopy of the late 1800s and early 1900s: s, p, d and f for l = 0, 1, 2 and 3, from the sharp, principal, diffuse and fundamental series of lines in alkali-metal spectra; beyond f the letters run on alphabetically, g, h, i, k, skipping j. A full label puts n in front, so 3d means n = 3 and l = 2. Every s orbital is spherical. A p subshell, meaning all the orbitals sharing one n and l, holds three orbitals, usually drawn as dumbbells along the x, y and z axes. A d subshell holds five: four four-leaf clovers and one, labelled z², a dumbbell threaded through a doughnut. An f subshell holds seven. The count is always 2l + 1, one per value of mₗ.

The shapes are carved by nodes, surfaces on which ψ, and so the probability density, is exactly zero. Angular nodes are planes or cones through the nucleus, exactly l of them: none for s, one plane for each p orbital, two planes or cones for each d orbital. Radial nodes are spheres centred on the nucleus, n − l − 1 of them, so the total is n − 1, just as the nth standing wave in chapter 11's box had n − 1 nodes. The 2s orbital has one radial node, a sphere of radius 2a₀, about 0.106 nm; 3s has two, near 1.90a₀ and 7.10a₀; the z² orbital's angular nodes are two cones at 54.7 degrees from the z axis. Only s orbitals have nonzero density at the nucleus.

Which picture is real? Label the states by mₗ and the 2p states with mₗ = +1 and −1 turn out to be not dumbbells but doughnuts circling the z axis, identical at every angle around it, because their wavefunctions only rotate in phase as you travel round. The familiar x and y dumbbells are superpositions, the sum and the difference, of those two states. A doughnut has no nodal plane, only a nodal line along the z axis, so the rule of l angular nodes counts the nodes of the real, dumbbell-style orbitals. Because the pair share one energy, every superposition is an equally valid stationary state, and choosing dumbbells or doughnuts is choosing a basis, exactly as in chapter 15. Chemists prefer the dumbbells, which point along bonds; physicists prefer definite mₗ when a magnetic field singles out the z axis.

When different states share one energy, the level is called degenerate, and hydrogen is spectacularly so. In level n, l runs from 0 to n − 1 and each l contributes 2l + 1 values of mₗ, so the total is 1 + 3 + 5 and so on up to 2n − 1, which always adds up to n². Level 1 has one state, 1s; level 2 has four, 2s plus three 2p; level 3 has nine and level 4 sixteen. Spin doubles every count, because each spatial state can hold the electron with mₛ = +½ or −½, giving 2n²: 2, 8, 18 and 32.

Degeneracy is the fingerprint of symmetry. The mₗ degeneracy follows from rotational symmetry: space has no preferred direction, so tilting an orbital cannot change its energy, in any central potential. The l degeneracy, which lets 2s and 2p share an energy, requires the exact 1/r Coulomb potential. It is often called accidental, but it springs from the hidden symmetry that keeps planetary orbits closed ellipses, the one Pauli exploited; in 1935 Vladimir Fock showed it to be a rotational symmetry in four dimensions. Distort the 1/r law, as the extra electrons of heavier atoms do, and the l degeneracy breaks; apply a magnetic field, which picks out a direction, and the mₗ degeneracy breaks too. Broken degeneracies appear as split spectral lines.

Eₙ = −13.6 eV/n² ignores spin and relativity, and nature is not quite that simple. In 1887 Albert Michelson and Edward Morley, examining hydrogen's red line with an interferometer, found that it is a close doublet. In 1916 Arnold Sommerfeld reproduced the splitting by adding relativity to Bohr's orbits, introducing the fine-structure constant α ≈ 1/137, the strength of electromagnetism, though partly by luck, since spin was unknown. Fine-structure splittings are smaller than the level energies by roughly α², about 5×10⁻⁵. They have two main sources: relativity, since the ground-state electron's typical speed is about 1/137 of the speed of light, and spin-orbit coupling, the interaction between the electron's spin magnetism and the magnetic field it feels from the proton's relative motion. Paul Dirac's relativistic equation of 1928 contains both.

A magnetic field cracks the pattern more visibly. In 1896 at Leiden, Pieter Zeeman put a sodium flame between the poles of an electromagnet of about one tesla and saw its yellow lines broaden; sharper sources resolved the broadening into separate components. Hendrik Lorentz explained the effect with his theory of electrons, and the two shared the 1902 Nobel Prize in Physics. In quantum terms, an electron with orbital angular momentum is a tiny magnet. A field B along z shifts each state's energy by mₗ times μ_B times B, where the Bohr magneton, μ_B = eħ/2mₑ, is about 5.79×10⁻⁵ electron-volts per tesla, so a level with a given l splits into 2l + 1 equally spaced sublevels: hence the name magnetic quantum number.

Yet many lines refused to play along. Beginning with Thomas Preston in Dublin in 1897, spectroscopists found more complicated patterns; sodium's two yellow lines split into four and six components, not the three that orbital magnetism alone allows. This anomalous Zeeman effect began to make sense in 1925, when two Leiden graduate students, George Uhlenbeck and Samuel Goudsmit, proposed that the electron carries intrinsic angular momentum, spin, with components of ±ħ/2 and twice the magnetism its angular momentum alone would suggest. Spin explained the anomalous patterns and, with relativity, the fine structure. Hydrogen's cracks were clues.

Hydrogen is the only neutral atom whose Schrödinger equation can be solved exactly, yet its solutions organize every atom. In a many-electron atom each electron feels the nucleus plus the averaged charge of the others, which is still roughly spherical, so the angular story carries over unchanged: the same quantum numbers, shapes, orientations and node counts. What changes is the radial story. Inner electrons shield the nucleus, so the potential is no longer a pure 1/r, the l degeneracy is lost, and within a shell s lies below p, which lies below d.

Now add the Pauli exclusion principle of chapter 18: no two electrons in an atom may share all four quantum numbers. Each spatial orbital then takes at most two electrons, one of each spin, and a shell can hold 2n² electrons: 2, 8, 18 and 32. Those numbers are written across the periodic table, whose rows hold 2, 8, 8, 18, 18, 32 and 32 elements; the repeats and reshuffles come from the broken l degeneracy, which lets 4s fill before 3d. Hydrogen supplies the seating chart; chapter 18 supplies the seating rules. Chemistry, from the angles of molecules to the colours of gemstones, is a negotiation among these shapes.