In theory, is it possible to simultaneously measure the angular momentum along $x$ and $z$ axis on the equator of a hydrogen atom?

The angular momentum operators $$L_x = yp_z-zp_y$$, $$L_x = zp_x-xp_z$$, $$L_x = xp_y-yp_x$$ do not commute. In fact, $$[L_x,L_y] = i\hbar L_z$$. We also know that $$L_z |nlm\rangle = \hbar m$$. So what if we have an eigenstate $$\psi = |100\rangle$$, this will make $$L_z \psi = 0$$. For this state, can we measure $$L_x$$ and $$L_y$$ simultaneously, in theory?

• "$\psi = |100>$... For this state, can we measure $L_x$ and $L_y$ simultaneously, in theor?" Yes, they are also zero.
– hft
Commented Nov 10, 2022 at 2:44
• I think the title of this question has a typo. Commented Nov 10, 2022 at 22:26
• Um... Can we measure any two things simultaneously for a single object? I think the fact you have commutators here indicates you don't really mean simultaneously. Lx Lz is one then the other. Commented Nov 10, 2022 at 22:39
• @BobaFit Yes, for example you can measure the $x$ and $y$ coordinates of a particle simultaneously. Also note that, following standard notation in quantum mechanics, the product $L_x L_y$ refers to multiplication of linear operators on Hilbert space, and not a sequence of measurements. Commented Nov 10, 2022 at 23:14

$$L_x, L_y, L_z$$ do commute on the subspace of states where $$L^2=0$$. So, if you prepare a state that lies in this subspace, you can measure all three simultaneously. (All three must be zero). That subspace includes the state $$|100\rangle$$ that you asked about explicitly.