The 9j symbol is related to the coupling probability for four angular momenta.

It is valid when

- j
_{i}≥ 0 - | j
_{1}- j_{2}| ≤ j_{3}≤ j_{1}+ j_{2}("triangle relations") - | j
_{4}- j_{5}| ≤ j_{6}≤ j_{4}+ j_{5} - | j
_{7}- j_{8}| ≤ j_{9}≤ j_{7}+ j_{8} - | j
_{1}- j_{4}| ≤ j_{7}≤ j_{1}+ j_{4} - | j
_{2}- j_{5}| ≤ j_{8}≤ j_{2}+ j_{5} - | j
_{3}- j_{6}| ≤ j_{9}≤ j_{3}+ j_{6}

See any quantum physics textbook for details.

See also my 3j Symbol and 6j Symbol Calculators.

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