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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.

Contributed by: Peter Falloon

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

Clebsch–Gordan Coefficient (Wolfram MathWorld)

ClebschGordan (The Wolfram Functions Site)

"Clebsch-Gordan Coefficients" from the Wolfram Demonstrations Project
http://demonstrations.wolfram.com/ClebschGordanCoefficients/

Contributed by: Peter Falloon

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Clebsch-Gordan Coefficients

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