In many resources I have seen that incompatible observables cannot have a common eigenbasis set, but may share one or few eigen states. I followed the thread Can incompatible observables share an eigenvector?, where through a matrix example it is proven to be true.

I want to know whether this can actually be true for physical incompatible observables like position-momentum in a direction, or angular momentum in two perpendicular directions and so on. If it happens to be true then in that particular state, measurement of both will be precise and the uncertainty principles:

$$\Delta x\Delta p_x\ge\frac{\hbar}{2} \hspace{1cm}\text{or}\hspace{1cm}\Delta L_x\Delta L_y\ge\frac{\hbar}{2}$$

seem to be violated.

There may be non-commuting matrices that can share an eigenvector. But whether those matrices can represent physical observables in some basis is a point of concern. Looking for opinions.

In many resources I have seen that incompatible observables cannot have a common eigenbasis set, but may share one or few eigen states. I followed the thread Can incompatible observables share an eigenvector?, where through a matrix example it is proven to be true.

I want to know whether this can actually be true for physical incompatible observables like position-momentum in a direction, or angular momentum in two perpendicular directions and so on. If it happens to be true then in that particular state, measurement of both will be precise and the uncertainty principles:

$$\Delta x\Delta p_x\ge\frac{\hbar}{2} \hspace{1cm}\text{or}\hspace{1cm}\Delta L\Delta \theta\ge\frac{\hbar}{2}$$

seem to be violated.

There may be non-commuting matrices that can share an eigenvector. But whether those matrices can represent physical observables in some basis is a point of concern. Looking for opinions.

In many resources I have seen that incompatible observables cannot have a common eigen basis set, but may share one or few eigen states. I followed the thread Can incompatible observables share an eigenvector?, where through a matrix example it is proven to be true.

I want to know whether this can actually be true for physical incompatible observables like position-momentum in a direction, or angular momentum in two perpendicular directions and so on. If it happens to be true then in that particular state, measurement of both will be precise and the uncertainty principles:

$$\Delta x\Delta p_x\ge\frac{\hbar}{2} \hspace{1cm}\text{or}\hspace{1cm}\Delta L_x\Delta L_y\ge\frac{\hbar}{2}$$

seem to be violated.

There may be non-commuting matrices that can share an eigen vector. But whether those matrices can represent physical observables in some basis is a point of concern. Looking for opinions.

In many resources I have seen that incompatible observables cannot have a common eigenbasis set, but may share one or few eigen states. I followed the thread Can incompatible observables share an eigenvector?, where through a matrix example it is proven to be true.

I want to know whether this can actually be true for physical incompatible observables like position-momentum in a direction, or angular momentum in two perpendicular directions and so on. If it happens to be true then in that particular state, measurement of both will be precise and the uncertainty principles:

$$\Delta x\Delta p_x\ge\frac{\hbar}{2} \hspace{1cm}\text{or}\hspace{1cm}\Delta L_x\Delta L_y\ge\frac{\hbar}{2}$$

seem to be violated.

There may be non-commuting matrices that can share an eigenvector. But whether those matrices can represent physical observables in some basis is a point of concern. Looking for opinions.

In many resources I have seen that incompatible observables cannot have a common eigen basis set, but may share one or few eigen states. I followed the thread 'https://physics.stackexchange.com/questions/656029/can-incompatible-observables-share-an-eigenvector', where through a matrix example it is proven to be true.

I want to know whether this can actually be true for physical incompatible observables like position-momentum in a direction, or angular momentum in two perpendicular directions and so on. If it happens to be true then in that particular state, measurement of both will be precise and the uncertainty principles  $$\Delta x\Delta p_x\ge\frac{\hbar}{2} \hspace{1cm}\text{or}\hspace{1cm}\Delta L_x\Delta L_y\ge\frac{\hbar}{2}$$ seem to be violated.

There may be non-commuting matrices that can share an eigen vector. But whether those matrices can represent physical observables in some basis is a point of concern. Looking for opinions.

In many resources I have seen that incompatible observables cannot have a common eigen basis set, but may share one or few eigen states. I followed the thread Can incompatible observables share an eigenvector?, where through a matrix example it is proven to be true.

I want to know whether this can actually be true for physical incompatible observables like position-momentum in a direction, or angular momentum in two perpendicular directions and so on. If it happens to be true then in that particular state, measurement of both will be precise and the uncertainty principles:

$$\Delta x\Delta p_x\ge\frac{\hbar}{2} \hspace{1cm}\text{or}\hspace{1cm}\Delta L_x\Delta L_y\ge\frac{\hbar}{2}$$ 

seem to be violated.

There may be non-commuting matrices that can share an eigen vector. But whether those matrices can represent physical observables in some basis is a point of concern. Looking for opinions.