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 ${\left\vert j\:m\right\rangle}$ 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 ${\left\vert j\:m\right\rangle}$ 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.)