D.63 An­gu­lar mo­men­tum un­cer­tainty

Sup­pose that an eigen­state, call it , of is also an eigen­state of . Then must be zero, and the com­mu­ta­tor re­la­tions say that this is equiv­a­lent to 0, which makes also an eigen­vec­tor of , and with the eigen­value zero to boot. So the an­gu­lar mo­men­tum in the -​di­rec­tion must be zero. Re­peat­ing the same ar­gu­ment us­ing the and com­mu­ta­tor pairs shows that the an­gu­lar mo­men­tum in the other two di­rec­tions is zero too. So there is no an­gu­lar mo­men­tum at all, is an state.