Hydrogen Atom: Fine Structure of Energy Levels

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

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

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



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 .

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.