I Angular Momentum Vector and Its Magnitude

[hokhani]

TL;DR

The relation between the quantized values of xyz components of angular momentum and its magnitude

For the quantum state $|l,m\rangle= |2,0\rangle$ the z-component of angular momentum is zero and $|L^2|=6 \hbar^2$. According to uncertainty it is impossible to determine the values of $L_x, L_y, L_z$ simultaneously. However, we know that $L_x$ and $L_y$, like $L_z$, get the values $(-2,-1,0,1,2) \hbar$. In other words, for the state $|2,0\rangle$ we have $\vec{L}=(L_x, L_y,0)$ with $L_x$ and $L_y$ one of the values $(-2,-1,0,1,2) \hbar$. But none of these configurations gives $|L^2|=6\hbar^2$!? I would appreciate if help me with this problem.

 

[PeroK]

Well, exactly! There is no realistic configuration that satisfies the requirements.

I'm suprised this example isn't shown more often. It's a good example of why QM isn't just classical mechanics with a bit of uncertainty thrown in. Spin values really, truly do not have well-defined values in a non-eigenstate.