It is known that the different spatial components of the angular momentum don't commute with each other.
$$
[L_x,L_y] \propto L_z \\
[L_y,L_z] \propto L_x \\
[L_z,L_x] \propto L_y
$$
Also it is known that the uncertainty depends on the commutator.
$$
\Delta L_x \Delta L_y \geq \frac{1}{2} \lvert \langle[L_x,L_y]\rangle\rvert
$$
This makes me wonder. Knowing, say, $L_z$, how does one put a number on the uncertainty in $L_y$?
For 1D it is straightforward.
$$
\Delta p = \frac{\hbar}{\Delta x}
$$
I can figure out $\Delta x$, width of the slit, for example, and then know the uncertainty in $p$.
However, to determine the uncertainty in $L_y$ , I need to know $L_x$, but because I know $L_z$ there is uncertainty associated with knowing $L_x$ proportional to $L_y$, since $
[L_z,L_x] \propto L_y
$. But I don't know $L_y$!
How would you figure this out?
Best Answer
For the angular momentum there is no lower bound for the product $(\Delta L_a)_\psi (\Delta L_b)_\psi$ differently from $x$ and $p$. Indeed there are states $\psi$ such that $L_a\psi =0$ for $a=x,y,z$ simultaneously. I am referring to the states with $L^2\psi =0$ which imply $$(\Delta L_a)_\psi (\Delta L_b)_\psi=0$$ So you cannot write something like $$ (\Delta L_x)_\psi (\Delta L_y)_\psi \geq k$$ for any constant $k>0$ independent from the state $\psi$ as it happens for $x$ and $p$. From this example you can see how, generally speaking, there exist simultaneous eigenstates of incompatible observables, even if you cannot find an orthonormal (Hilbert) basis of these vectors.