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Question

# (a) Derive an expression for the angle between the vector representing angular momentum l with z-component $m_{1}=+l$ (that is, its maximum value) and the z-axis. What is this angle for l = 1 and for l = 5? (b) Derive an expression for the angle between the vector representing angular momentum l with z-component $m_{1}=+l$ and the z-axis. What value does this angle take in the limit that l becomes very large? Interpret your result in the light of the correspondence principle.

Solution

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$\textbf{(a)}$

Expression for the angle between the vector representing angular momentum $l$ with z - component $m_l = + l$ (i.e. its maximum value) can be obtained from equation that defines angular momentum:

$\begin{equation}|\vec{l}| = \sqrt{l (l + 1)}\hbar\end{equation}$

and its projection on z - axis:

$\begin{equation}l_z = m_l \hbar, \end{equation}$

where $l$ is orbital quantum number ($l = 0,1,2,...n-1$; n is principal quantum number), $m_l$ magnetic orbital quantum number ($m_l = 0, \pm 1, \pm 2, ....,\pm l$), $\hbar = \dfrac{h}{2 \pi}$ ($h$ is Planck constant).

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