D.63 An­gu­lar mo­men­tum un­cer­tainty

Sup­pose that an eigen­state, call it $\big\vert m\big\rangle$, of ${\widehat J}_z$ is also an eigen­state of ${\widehat J}_x$. Then $[{\widehat J}_z,{\widehat J}_x]\big\vert m\big\rangle$ must be zero, and the com­mu­ta­tor re­la­tions say that this is equiv­a­lent to ${\widehat J}_y\big\vert m\big\rangle$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, which makes $\big\vert m\big\rangle$ also an eigen­vec­tor of ${\widehat J}_y$, and with the eigen­value zero to boot. So the an­gu­lar mo­men­tum in the $y$-​di­rec­tion must be zero. Re­peat­ing the same ar­gu­ment us­ing the $[{\widehat J}_x,{\widehat J}_y]$ and $[{\widehat J}_y,{\widehat J}_z]$ com­mu­ta­tor pairs shows that the an­gu­lar mo­men­tum in the other two di­rec­tions is zero too. So there is no an­gu­lar mo­men­tum at all, $\big\vert m\big\rangle$ is an ${\left\vert\:0\right\rangle}$ state.