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Highest-weight states in angular momentum addition

With

J = j_{1} + j_{2}

, the maximal state is the product

∣ J, J ⟩ = ∣ j_{1}, j_{1} ⟩ ∣ j_{2}, j_{2} ⟩

. Lowering it once gives

∣ J, J - 1 ⟩ = \frac{j _{1}}{J} ∣ j_{1}, j_{1} - 1 ⟩ ∣ j_{2}, j_{2} ⟩ + \frac{j _{2}}{J} ∣ j_{1}, j_{1} ⟩ ∣ j_{2}, j_{2} - 1 ⟩ .

The orthogonal combination is the highest-weight state

∣ J - 1, J - 1 ⟩

.

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