Why can't we measure the different components of angular momentum simultaneously?

Question

Why can't we measure the different components of angular momentum simultaneously? I want a physical explanation. (I know that they don't commute with each other).

Answer 1

Imagine that you define the states $\{|l \rangle\}$ as the eigenstates of the operator $L_z$, that is $L_z|l\rangle = l |l\rangle$. Any operator $O$ that commutes with $L_z$ has the same eigenstates, indeed

$$ L_z (O|l\rangle) = O(L_z|l\rangle) = O(l |l\rangle) = l (O|l\rangle) $$

that means that $O|l\rangle$ must be an eigenstate of $L_z$ with eigenvalue $l$.

Now, if on the other hand $O$ does not commute with $L_z$, the result of $O|l_z\rangle$ will be any other state, in general a linear combination of $\{|l\rangle\}$ but . This means that

$$ O |l\rangle = \sum_{l'} c_{l'}|l'\rangle \not= l |l\rangle $$

With this in mind, imagine a system you prepared in the eigenstate $l$ of $L_z$, and the next two situations

• Measure $L_z$ and then $L_x$: In the first step you get $l$ and the system stays in the state $|l\rangle$. Then, when you measure $L_x$, the wavefunction will collapse to any of the eigenstates of $L_x$ which are different from $|l\rangle$

• Measure $L_x$ and then $L_z$: In the first step the measure will leave the system in any of the eigenstates of $L_x$, and then when you measure $L_z$ it will give you any of the eigenstates of $L_z$, different from $|l\rangle$

So you measuring these two observables in different order, will yield different answers, in other words, you cannot hope to measure both simultaneously and get the same answer