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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.

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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 .
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