Do Incompatible Observables Have Common Eigenfunctions?

[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)?

 

[quantumdude]

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 L_x, L_y, and L_z is the spherical harmonic Y₀₀, with a zero eigenvalue.

 

[R0man]

One example of incompatible observables that have common eigenfunctions but incomplete set of eigenfunctions is the position and momentum operators in quantum mechanics. The position operator, denoted as x, and the momentum operator, denoted as p, do not commute with each other. This means that there is no single set of common eigenfunctions that can simultaneously diagonalize both operators. However, there are certain wavefunctions that are eigenfunctions of both x and p, such as the Gaussian wavefunction.

But even though these eigenfunctions are shared by both operators, they do not span the entire Hilbert Space. This is because there are other wavefunctions that are eigenfunctions of x or p, but not both. For example, a plane wave is an eigenfunction of the momentum operator but not the position operator. Therefore, the set of common eigenfunctions of x and p is incomplete and does not span the entire Hilbert Space.

This shows that incompatible observables, while having some common eigenfunctions, do not have a complete set of common eigenfunctions. This is due to the fact that their operators do not commute, leading to a lack of shared eigenfunctions and an incomplete set of eigenfunctions.