Compatibility of qunatum measurements

[sachi]

The question is "two operators A,B do not commute". Is it true that
([A,B]psi) does not equal zero for any state psi?"

I'm not sure if there are any cases in which a state psi could produce this result (except for the obvious one where psi = 0 and there is no particle).

Thanks

 

[Physics Monkey]

Here is a hint: think about the commutator $$[x^2 , p]$$. Can you find a state that gives zero when you hit it with the commutator even though commutator isn't identically zero?

 

[sachi]

I've tried applying the commutator to an arbitrary psi and i get i*h bar * 2x* psi. The only case where I can think of this equalling zero is when psi = 0, which is the obvious case of having no particle.

 

[Physics Monkey]

You can't think of any state that gives zero when you hit it with the position operator? Hint: think about position eigenstates.