Clebsch-Gordan Coefficients

This Demonstration illustrates the Clebsch–Gordan coefficients, , which give the coupling amplitudes between uncoupled and coupled representations of two angular momenta and . In the uncoupled representation, the components of each angular momentum, and , are known; in the coupled representation, the total (resultant) angular momentum and its component are known. The Clebsch–Gordan coefficients are only nonzero when and ; in the Demonstration we show these for . The graphs give a vectorial representation of each pair, showing the actual value together with all possible values.


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


In quantum mechanics, angular momentum is quantized in units of . The allowed values are specified by the quantum number ; for a given , the corresponding total angular momentum has value . In addition, one Cartesian component—conventionally the component—can also be specified, and can take on values where . The other two components cannot be specified individually, which is a manifestation of the uncertainty principle.
The Clebsch–Gordan coefficients arise in systems comprising two angular momenta, and . It is possible to define either states with well-defined individual components and (the uncoupled representation), or well-defined total angular momentum and its component (the coupled representation). Allowed values in the coupled representation are and . The amplitudes relating the two representations are the Clebsch–Gordan coefficients .
    • 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.

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 © 2018 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+