N.27 Magnitude of components of vectors

You might wonder whether the fact that the square components of angular momentum must be less than total square angular momentum still applies in the quantum case. After all, those components do not exist at the same time. But it does not make a difference: just evaluate them using expectation values. Since states $\big\vert j\:m\big\rangle $ are eigenstates, the expectation value of total square angular momentum is the actual value, and so is the square angular momentum in the $z$-​direction. And while the $\big\vert j\:m\big\rangle $ states are not eigenstates of ${\widehat J}_x$ and ${\widehat J}_y$, the expectation values of square Hermitian operators such as ${\widehat J}_x^2$ and ${\widehat J}_y^2$ is always positive anyway (as can be seen from writing it out in terms of the eigenstates of them.)