If for each value of the orbital quantum number $\ell$ there are $2\ell+1$ possible associated magnetic quantum numbers $m_\ell$, and they are interpreted as the only allowed orientations that the $L_z$ component of the angular momentum can adopt, what implication does this have in relation to the correspondence principle when $\ell\gg1$ and therefore also $2\ell+1\gg1$? More specifically, what is the physical interpretation (with respect to the correspondence principle) of the fact that the number of allowed orientations is very large?
The correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and large energies, quantum calculations must match classical calculations.
Then, if $\ell \to \infty $ we will have values for $m_\ell$ in $(-\infty, \infty)$ But I don't know how to interpret this physically and see how it coincides with the classical results.
Best Answer
In quantum mechanics angular momentum is quantized like this: $$L^2=\ell(\ell+1)\hbar^2$$ $$L_z=m_\ell\hbar$$
Semiclassically the angle $\theta$ between $\vec{L}$ and the $z$-axis is given by $$\cos\theta=\frac{L_z}{L} =\frac{m_\ell}{\sqrt{\ell(\ell+1)}}\approx\frac{m_\ell}{\ell}$$
This means, when $\ell$ is large, then this angle $\theta$ can change in very small steps. And for $\ell\to\infty$ the angle can take every real value between $0°$ and $180°$.