Clebsch-Gordan Coefficients

component angular momenta

j

1

1/2

1

3/2

2

5/2

3

j

2

1/2

1

3/2

2

5/2

3

uncoupled representation

m

1

-1/2

1/2

m

2

-1/2

1/2

coupled representation (M 

m

1

+

m

2

)

J

1



1

2

1

2

1

2

1

2

1

2

1

2

111

j 1 = 1/2, m 1 = 1/2	j 2 = 1/2, m 2 = 1/2	J = 1, M = 1

This Demonstration illustrates the Clebsch–Gordan coefficients,

〈

j

1

j

2

m

1

m

2



j

1

j

2

JM〉

, which give the coupling amplitudes between uncoupled and coupled representations of two angular momenta

j

1

and

j

2

. In the uncoupled representation, the

z

components of each angular momentum,

m

1

and

m

2

, are known; in the coupled representation, the total (resultant) angular momentum

J

and its

z

component

M

are known. The Clebsch–Gordan coefficients are only nonzero when

M

m

1

+

m

2

and

max(M,

j

1

-

j

2

)≤J≤

j

1

+

j

2

; in the Demonstration we show these for

j

1

,

j

2

≤3

. The graphs give a vectorial representation of each

(j,m)

pair, showing the actual value together with all possible

m

values.