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.