Polynomial termination fixes the hydrogen energy. In the ideal nonrelativistic Coulomb problem, all states with the same principal quantum number share one energy even when their l and m values differ.
The continuum begins at E = 0; 13.6 eV is the ground-state ionization energy.
A hydrogen state is specified by the triplet \((n,l,m)\) before spin is included.
For fixed n, count all m states in every allowed l subshell:
Including the two spin states doubles this to \(2n^2\).
Bound-bound transitions create discrete spectral lines; a final energy at or above zero corresponds to ionization.
The decay constant is related to energy by \(E=-\hbar^2\kappa^2/(2m)\). The series termination condition also fixes \(\kappa=1/(na_0)\). Combining them gives \(E_n=-\hbar^2/(2ma_0^2n^2)=-E_0/n^2\). The derivation shows why energy depends on n but not separately on l or m in the ideal Coulomb problem.
For each l there are \(2l+1\) values of m. Summing from l = 0 through n - 1 gives \(1+3+5+\cdots+(2n-1)=n^2\). For n = 3, the 3s, 3p and 3d subshells contain 1, 3 and 5 orbital states, totaling 9. Electron spin doubles the capacity but is not part of the spatial Schrödinger solution developed here.
Use the following chain when working with Hydrogen spectrum and degeneracy. 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: Principal quantum number, Bohr energy scale, subshells and the n-squared degeneracy of the ideal atom. The final answer should explicitly connect this objective to at least one equation on the page and one experimentally meaningful prediction.
Spherical symmetry explains m degeneracy, but the additional l degeneracy of the Coulomb problem is stronger. Fine structure and external fields later lift parts of this ideal degeneracy.
This page is designed to support short guided exercises on: Principal quantum number, Bohr energy scale, subshells and the n-squared degeneracy of the ideal atom.
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.