In my teacher's notes there is a discussion of the Hamiltonian for a central force field with potential $V(r)$.
The Hamiltonian is formulated in spherical polar coordinates:
$$H=\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2\sin^2\theta}+\frac{p_\theta^2}{2mr^2}+V(r)$$
Then the conservation of $L_z$ is trivial, because $\phi$ is cyclic. However it is asserted, as if it were evident, that $L^2$ is also conserved. One consequence is that the Hamiltonian can be written in the form $H=\frac{p_r^2}{2m}+\frac{L^2}{2mr^2}+V(r)$.
Now, I know that for any radial force it is trivial to prove that $\vec L$ is conserved, but how can you prove that $L^2$ is conserved just from looking at the previous Hamiltonian? Moreover, does it suffices to prove that $L^2$ and $L_z$ are conserved in order to find that all the three components of $\vec L$ are conserved?
I'm a bit confused, so any help is appreciated
Best Answer
By comparing the two expressions for $H$ you can infer an expression for $L^2$, which you need to prove, by transforming the standard expression for $L^2$ to spherical polar coordinates.
Conservation of $L^2$ and $L_z$ is enough to infer the spectrum, but one cannot deduce from it conservation of $L$.