# Hydrogen Atom: Fine Structure of Energy Levels

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

The energy levels of the hydrogen atom , taking account only of the Coulomb interaction between the electron and proton, are shown on the left. Perturbed energy levels , also including spin-orbit interaction and relativistic corrections, produce the so-called fine structure, as shown on the right.

Contributed by: Lukás Rafaj (May 2017)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Atomic structure, specifically for the hydrogen atom, is determined principally by Coulomb interactions among electrons and the nucleus. This leads to the unperturbed energy . There also exist smaller contributions to the energy, most notably from spin-orbit interactions. These are interactions between orbital and spin magnetic moments of the electron, represented by the Hamiltonian

,

where and are the orbital and spin angular momenta, respectively. A secondary perturbation comes from relativistic corrections to electron kinetic energy, represented by a term in the Hamiltonian of the form

.

The total fine structure is then represented by the perturbation

,

with a first-order energy correction

.

This gives the total energy of the state , , to first order in perturbation theory,

,

where is the principal quantum number, is the total electronic angular momentum quantum number and is the fine-structure constant .

## Permanent Citation