# Why is $L_z$ operator more important the $L_x$ or $L_y$ operators?

When we talk about orbital angular momentum, we always use L_z but never talk about L_x or L-y. Why is that?

• Also, notice that what you call $L_z$ is $L_x$ for some other observers, so it doesn't really matter. – gented Jun 3 at 8:39

There is nothing special about $$L_{z}$$. In many cases, it is literally just an arbitrary choice. You need to find a complete set of commuting observables (CSCO) to completely classify the states of a theory; when angular momentum is involved, it means that you need to choose one of the projections of $$\vec{L}$$ to use in your CSCO, and it is conventional to select $$L_{z}$$.
In other cases, the choice of $$L_{z}$$ is not quite so arbitrary, but it is, rather, an outgrowth of a different choice of convention. When there is an external magnetic field, it is conventional to denote the magnetic field direction as the $$z$$-direction: $$\vec{B}=B\hat{z}$$. In the presence of such a field, the energy levels of different angular momentum states are split. However, the eigenstates of $$L_{z}$$ are still eigenstates of the Zeeman effect Hamiltonian $$-\vec{\mu}\cdot\vec{B}$$, while the eigenstates of $$L_{x}$$ or $$L_{y}$$ are not. So the choice of the $$z$$-direction for the magnetic field make $$L_{z}$$ the natural angular momentum component to use.