AT · Atomic spectra and selection rules

Not every energy difference produces a strong electric-dipole 电偶极 line.

Prerequisites: 23, 40, 41.

• Relate atomic energies to spectral lines
• Apply angular-momentum quantum numbers
• Distinguish allowed and forbidden electric-dipole transitions

For hydrogen-like one-electron ions, the bound energies scale as −13.6 Z²/n² eV in the simplest nonrelativistic model. Emission requires a downward transition and positive photon energy.

Orbital angular momentum has l=0,…,n−1 and m_l=−l,…,l. Electron spin adds s=1/2; total angular momentum combines orbital and spin contributions.

Electric-dipole selection rules include Δl=±1 and Δm=0,±1. These rules concern a specific transition mechanism; forbidden does not mean impossible by every mechanism.

Fine structure, spin–orbit effects and Zeeman splitting 塞曼分裂 refine the simplest hydrogen picture. In many-electron atoms, shielding and electron interactions invalidate direct hydrogen-energy substitution.

For a nonrelativistic one-electron ion, neglecting fine structure,

For Z = 2, an emission from n = 3 to n = 2 gives positive photon energy: An energy difference alone does not guarantee an electric-dipole transition; the participating orbital states must satisfy the selection rules.

Do not make the final-state energy minus initial-state energy negative for an emitted photon.

selection rule 选择定则: A restriction on transitions for a specified interaction mechanism.

Zeeman splitting: Magnetic-field splitting of atomic energy levels.