When two operators switch have a complete set of simultaneous eigenstates, are the simultaneous eigenstates that are part of this complete set all the simultaneous eigenstates of the two existing operators?
Could there be other simultaneous eigenstates not belonging to this set?
For example, when we're searching for the simultaneous eigenstates of the commuting operators $H$, $L^2$ and $L_z$ in the problem of a particle inside a central field, we look for them among the spherical harmonics.
How can we be sure that simultaneous eigenstates of $H$, $L^2$ and $L_z$ are among simultaneous eigenstates of $L^2$ and $L_z$, namely spherical harmonics?
Is it not possible for there to be simultaneous eigenstates of $H$, $L^2$, $L_z$ outside the set of simultaneous eigenstates of $L^2$, $L_z$?
Best Answer
This question can be answered without anything quantum: the set of things that have properties $A$, $B$, and $C$ is always contained within the set of things that have properties $A$ and $B$, because anything with $A$, $B$, and $C$ must (by definition) have $A$ and $B$.
The converse is not true: something with properties $A$ and $B$ doesn't necessarily have all three properties $A$, $B$, and $C$.
To apply to your problem: anything that is a simultaneous eigenstate of $H$, $L^2$, and $L_z$ must by definition be an eigenstate of $L^2$ and $L_z$. Therefore it is impossible to have a simultaneous eigenstate of $H$, $L^2$, and $L_z$ outside the set of simultaneous eigenstates of $L^2$ and $L_z$.