Hydrogenic quantum numbers provide the language of shells and subshells used for multi-electron atoms. Electron spin, the exclusion principle and empirical energy ordering turn this language into the block structure of the periodic table.
For fixed l there are \(2l+1\) values of m. Including two spin states per orbital gives
| Subshell | l | Capacity |
|---|---|---|
| s | 0 | 2 |
| p | 1 | 6 |
| d | 2 | 10 |
| f | 3 | 14 |
Summing over l = 0 to n - 1 gives
The s, p, d and f blocks have widths 2, 6, 10 and 14 because those are their electron capacities. Periods follow the order in which subshells are filled.
Madelung ordering is a useful empirical rule, not an exact theorem for every atom.
For a subshell l, the allowed m values give \(2l+1\) spatial orbitals. The Pauli principle permits two electrons of opposite spin in each orbital, producing capacity \(2(2l+1)\). Summing all subshells in a hydrogenic shell gives \(2n^2\), explaining the familiar shell capacities before electron-electron interactions modify the energy ordering.
Hydrogen energy depends only on n, but in many-electron atoms screening and penetration make energy depend strongly on l. Low-l orbitals penetrate closer to the nucleus and often feel a larger effective charge. The empirical n + l rule summarizes the resulting order reasonably well. Exceptions occur because the true problem includes electron repulsion, exchange and relativistic effects, so the rule should guide rather than replace calculation.
Use the following chain when working with Hydrogen atom and the periodic table. First identify the symmetry, prepared state and observable being discussed; this determines which basis and quantum numbers are meaningful. Next write the operator or differential equation before substituting eigenvalues. Apply boundary conditions, commutators or normalization one step at a time, keeping dimensions and allowed quantum-number ranges visible. Only then simplify to the highlighted result. Finally translate the mathematics into a measurement statement: specify which outcomes are possible, which quantities remain uncertain, how degeneracy is counted, or where probability is concentrated. This sequence prevents a correct formula from becoming disconnected from the physical assumptions that justify it.
Self-check: Shells, subshells, orbital capacity, Madelung ordering and the quantum structure behind the periodic table. The final answer should explicitly connect this objective to at least one equation on the page and one experimentally meaningful prediction.
In hydrogen, all l values within one n shell are degenerate. Electron-electron interactions in many-electron atoms break that degeneracy, making subshell filling nontrivial.
This page is designed to support short guided exercises on: Shells, subshells, orbital capacity, Madelung ordering and the quantum structure behind the periodic table.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.