# Clebsch-Gordan Coefficients

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.

〈jjmmjjJM〉

1

2

1

2

1

2

j

1

j

2

z

m

1

m

2

J

z

M

Mm+m

1

2

maxM,j-j≤J≤j+j

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2

1

2

j,j≤3

1

2

(j,m)

m