incompatible observables

[kakarukeys]

Incompatible observables do not share a complete set of common eigenfunctions, because their operators do not commute.

It seems that, incompatible observables like x and p_x, S_x and S_y do not have any common eigenfunction at all.

Can anyone give a concrete example of a pair of incompatble observables that have common eigenfunctions but incomplete (do not span the Hilbert Space)?

[Tom Mattson]

Can anyone give a concrete example of a pair of incompatble observables that have common eigenfunctions but incomplete (do not span the Hilbert Space)?

The only example I can think of is the state of zero orbital angular momentum. The eigenfunction common to the incompatible operators Lx, Ly, and Lz is the spherical harmonic Y00, with a zero eigenvalue.