The normal triangle inequality continues to apply for expectation values in quantum mechanics.
The way to show that is, like other triangle inequality proofs, rather
curious: examine the combination of , not with
, but with an arbitrary multiple of
If you multiply out, you get
Note that this derivation does not use any properties specific to angular momentum and does not require the simultaneous existence of the components. With a bit of messing around, the azimuthal quantum number relation can be derived from it if a unique value for exists; the key is to recognize that where is an increasing function of that stays below , and the values must be half integers. This derivation is not as elegant as using the ladder operators, but the result is the same.