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Mathematics for Physics

122 Hydrogen Energy Levels

A discrete set of allowed energies of the hydrogen atom that is used to compute the bound-state spectrum, where a bound state is a state with negative energy confined by the Coulomb potential.

The \(1/n^{2}\) hydrogen spectrum. The allowed energies of hydrogen fall as the inverse square of the principal quantum number. This principle is used to write the entire discrete spectrum from one formula.

The hydrogen energy levels are

\[ E_{n} = -\dfrac{13.6\,\mathrm{eV}}{n^{2}} = -\dfrac{\mu e^{4}}{32\pi^{2}\epsilon_{0}^{2}\hbar^{2}}\dfrac{1}{n^{2}} \]

where

  • \(E_{n}\) is the energy of level \(n\).
  • \(n\) is the principal quantum number.
  • \(\mu\) is the reduced mass of the electron-proton pair.
  • \(e\) is the elementary charge.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(\hbar\) is the reduced Planck constant.

The ground-state energy. The ground state is \(n=1\) with \(E_{1}=-13.6\,\mathrm{eV}\). This principle is used to set the ionization energy of hydrogen.

Coulomb degeneracy. Levels with the same \(n\) share the same energy in the pure Coulomb problem. Degeneracy is the sharing of one energy by several states. This principle is used to count the \(n^{2}\) Coulomb states of a given \(n\) before spin.

The connection to the Rydberg formula. A jump from \(n_{2}\) to \(n_{1}\) emits a photon whose wavenumber is given by the Rydberg formula. This principle is used to convert the level diagram into spectral series.

122.1 References

  1. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. §4.2 — hydrogen atom energies.
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — hydrogen spectrum.