The Periodic Table and Atomic Spectra

Hydrogen is a one-electron problem solved exactly. Every heavier atom adds electrons that repel one another, and the Schrödinger equation no longer separates. Two ideas carry the analysis through. Electron-electron repulsion shifts the energies of the hydrogenic states, breaking the degeneracy in $\ell$; and the indistinguishability of electrons imposes a rule — the exclusion principle — that decides how many electrons a shell can hold. Together they generate the periodic table and the optical spectra of the elements.

Identical particles and exchange symmetry

For two particles the Schrödinger equation involves both coordinates. If the particles do not interact, solutions are products of single-particle states,

but this form distinguishes the particles: swapping the labels gives a different function. Identical particles cannot be told apart, so the probability density must be unchanged under exchange:¹

This forces $\psi$ to be either symmetric or antisymmetric under the swap. Neither the bare product nor a single term qualifies; the acceptable combinations are

Particles with symmetric wave functions — $\alpha$ particles, deuterons, photons, mesons — do not obey the exclusion principle; they may pile into one state. The antisymmetry of electrons is what gives matter its shell structure.

Filling the shells

Neglecting electron-electron interaction, each electron in a heavier atom occupies a hydrogen-like state labeled by $n,\ell ,m_{\ell },m_s$. The energy rises with both $n$ and $\ell$. The $\ell$ dependence is new: it comes from penetration and shielding.² Inner electrons screen the nuclear charge, so an outer electron sees an effective charge $Z_{\text{eff}}$ between $1$ and $Z$. A low-$\ell$ orbital, with its inner probability bump (recall the 2s radial distribution), penetrates the screening cloud, sees more nuclear charge, and binds more tightly. For a given $n$, energy therefore increases with $\ell$: $3s<3p<3d$.

The shift is large enough to reorder shells. Beyond argon the $4s$ level drops below $3d$, so potassium and calcium fill $4s$ before the $3d$ subshell begins.

Walking up $Z$ and applying the exclusion principle gives the electron configurations:²

• Helium ($Z=2$): $1s^2$. Two electrons of opposite spin fill the K shell; the total spin is zero. The high ionization energy makes it inert.
• Lithium ($Z=3$): $1s^{2}2s$. The third electron cannot join the full K shell, so it enters $n=2$, choosing $2s$ over $2p$ because $2s$ penetrates the core.
• Beryllium ($Z=4$): $1s^{2}2s^2$. The exclusion principle pairs the fourth electron with the second $2s$ electron, opposite spin.
• Boron to neon ($Z=5$ to $10$): the $2p$ subshell fills its six slots ($3$ values of $m_{\ell }$, $2$ of $m_s$), ending at neon, $1s^{2}2s^{2}2p^6$, inert.
• Sodium to argon ($Z=11$ to $18$): the $3s$ then $3p$ subshells fill, ending at argon, $1s^{2}2s^{2}2p^{6}3s^{2}3p^6$.

The recurrence of a filled outer shell is the periodicity. Elements sort into blocks by which subshell is filling.

The clearest signature of shell closing is the first ionization energy, the energy to remove the outermost electron. It climbs across a period and drops sharply at each alkali metal, where a new shell starts one electron far from a screened core. The peaks fall at the noble gases $Z=2,10,18,36,54,86$.

Worked example: the effective charge in lithium

The measured first ionization energy of lithium is $5.39$ eV, and its outer electron has $n=2$. Modeling it as a hydrogenic electron of effective charge $Z_{\text{eff}}$,

gives

The $2s$ electron sees an effective charge slightly above $1$: the two K-shell electrons screen most of the $+3$ nucleus, but the $2s$ penetration adds a little.²

Atomic spectra

Optical spectra come from transitions of the outermost electron; core transitions land in the ultraviolet and X-ray. The alkali metals — Li, Na, K, Rb, Cs — behave almost hydrogenically, one electron over a closed core, so their spectra resemble hydrogen's. Two rules govern which transitions appear:³

In sodium the outer $3s$ electron excites to $3p$, about $2.1$ eV up. Spin-orbit coupling splits $3p$ into ${}^2{P}_{3/2}$ and ${}^2{P}_{1/2}$, separated by only about $0.002$ eV. The two transitions to the ground state produce the famous sodium yellow doublet, the color of sodium street lamps:

The Zeeman effect

An external magnetic field $\vec{B}$ supplies the preferred direction that free space lacks. The total angular momentum $\vec{J}$ quantizes along $\vec{B}$, and a level of quantum number $j$ splits into $2j+1$ sublevels, one for each $m_j$, with energy shift proportional to $m_j$. Split levels split the spectral lines. In the simplest (normal) case a single line becomes three, governed by $\Delta m=0,\pm 1$.³ Zeeman and Lorentz shared the Nobel Prize for the discovery and its explanation.

The atomic physics of this module is thus a chain of consequences from the Schrödinger equation: three quantum numbers from the three coordinates, a fourth from electron spin, a rule of antisymmetry that stacks electrons into shells, and selection rules that decide which lines an atom emits. The same shell idea, with protons and neutrons in place of electrons, reappears in the shell model of the nucleus.