I am reading the first chapter of Sakurai's Modern QM and from pages 30 and 32 respectively, I understand that(adsbygoogle = window.adsbygoogle || []).push({});

(i) If [A,B]=0, then they share the same set of eigenstates.

(ii) Conversely, if two operators have the same eigenstates, then they commute.

But we know that [itex][L^2,L_z]=0[/itex], [itex][L^2,L_x]=0[/itex] and [itex][L_z,L_x]\neq 0[/itex].

From the first equality, and (i) I gather that L² and L_z have the same eigenkets. From (i) and the second equality, I get that L² and L_x have the same eigenkets. So L_z and L_x have the same eigenkets. Hence, by (ii), they commute, which contradicts [itex][L_z,L_x]\neq 0[/itex].

Where did I go wrong??

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# Commutation relations trouble (basic)

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