Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-27T01:20:59.679Z Has data issue: false hasContentIssue false

12 - 9 j-Coefficients and Higher

Published online by Cambridge University Press:  30 September 2020

Tom H. Koornwinder
Affiliation:
Universiteit van Amsterdam
Jasper V. Stokman
Affiliation:
Universiteit van Amsterdam
Get access

Summary

The su(2) 3j-coefficients (or symbols) and higher ones as 6j and 9j play a crucial role in various physical applications dealing with the quantization of angular momentum. In this chapter, the hypergeometric expressions for these coefficients and their relations to discrete orthogonal polynomials are emphasized. We give a short summary of the relevant class of representations of the Lie algebra su(2), and discuss their tensor product. In the tensor product decomposition, the important Clebsch-Gordan coefficients appear. 3j-Coefficients are proportional to these Clebsch-Gordan coefficients. We give some useful expressions (as hypergeometric series) and their relation to Hahn polynomials. Next, the tensor product of three representations is considered, and the relevant Racah coefficients (or 6j-coefficients) are defined. The explicit expression of a Racah coefficient as a hypergeometric series of type 4F3 and the connection with Racah polynomials and their orthogonality is given.9j-Coefficients are then defined in the context of the tensor product of four representations. They are related to a discrete orthogonal polynomial in two variables. Finally, we consider the tensor product of (n+1) representations and generalized recoupling coefficients or 3nj-coefficients, determined by two binary coupling schemes.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×