ATM-009 · Atoms and molecules Sample material

One-electron atoms

The Schrödinger equation for an electron in a centrally symmetric field: separation of variables in spherical coordinates, the radial equation, eigenvalues and eigenfunctions of hydrogen-like atoms.

Estimated time: 3 h 3 activities

Learning objective

Explain the properties of a centrally symmetric potential and the conservation of angular momentum.
Learn to separate the Schrödinger equation in spherical coordinates into angular and radial parts.
Understand the meaning of the radial equation and the effective potential.
Calculate the energy levels and radial distributions of hydrogen-like ions.

Expected outcomes

Calculates the levels and spectral lines of H, He⁺, Li²⁺ with $E_n = -13.6\,Z^2/n^2$ eV.
Knows the ranges of the quantum numbers $n$, $l$, $m$ and the level degeneracy ($n^2$).
Finds the most probable radius from the radial probability density $P(r) = r^2R^2$.
Takes the reduced-mass correction into account.

Interactive modelRadial probability distribution in hydrogen

Activities

Lecture
Lecture · 1 h 20 min · required
One-electron atoms. The potential of a centrally symmetric field. Energy levels. Quantum numbers.
Practice · 1 h 20 min · required
Final test
Test · 20 min · required · pass ≥ 70%

Completion conditions

  • Lecture
  • One-electron atoms. The potential of a centrally symmetric field. Energy levels. Quantum numbers.
  • Final test — at least 70%