# How to calculate Clebsch-Gordon coefficients

1. ### Avijeet

40
Can anybody tell me how to calculate Clebsch-Gordon coefficients?
I see a table given for the coefficients in some books (Griffiths p200), but it is not clear how to read the table.
Any help would be appreciated.

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3. ### SpectraCat

1,395
Well, I never use the Clebsch-Gordon coefficients directly in calculations ... I use Wigner 3-j symbols instead.(see http://en.wikipedia.org/wiki/3-jm_symbol) The CG coeffs are useful because they are directly related to the angular momentum coupling equations, but the Wigner symbols are much more intuitive to use in calculations, and have useful symmetry properties as well. I am not going to give a detailed re-hashing of the CG coeffs here ... the treatment in Griffiths is good .. you could also try Zare's "Angular Momentum", which gives a more thorough description IMO. If you have specific questions, please ask them .. in the meantime I can give the following descriptive summaries that may prove helpful.

CG coeffs are the scalar products of the description between the uncoupled and coupled representations for two angular momentum vectors j1 and j2. In usual notation, $<j_1m_1j_2m_2|j_1j_2JM>$, the uncoupled representation is on the left, where the z-projections of the two angular momenta are considered separately. The coupled representation is on the right, where the two angular momenta are first added together to give total angular momentum (J), and it's projection on the z-axis (M). Remember there are multiple ways that two angular momenta can be added ... that is the reason we need the CG coefficients in the first place.

The formula for calculating a CG coefficient can be found here: http://en.wikipedia.org/wiki/Table_of_Clebsch-Gordan_coefficients#Formulation.