11428

Hydrogen Atom: Fine Structure of Energy Levels

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.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

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 .
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+