Schrödinger's Equation and the Hydrogen Atom — Question 7

Question 7

Orbital angular momentum has a fixed magnitude and discrete axial projections

Treat hydrogen with a fixed nucleus, a nonrelativistic Coulomb potential and no spin, fine structure or external field. Write the Bohr length as a. A state has $n=3$, $\ell=2$ and $m_\ell=2$. Use $L^2=\ell(\ell+1)\hbar^2$ and $L_z=m_\ell\hbar$.

1. Find the angular-momentum magnitude and z component.

2. List all possible z components for this $\ell$ and find the largest ratio $|L_z|/|L|$.

Question 7 - Solution

Orbital angular momentum has a fixed magnitude and discrete axial projections

(a) Method and calculation

$$\boxed{|L|=\sqrt6\,\hbar},\qquad\boxed{L_z=2\hbar}.$$$$\boxed{|L|^2-L_z^2=2\hbar^2}.$$

(b) Interpretation and further calculation

$$\boxed{L_z\in\{-2,-1,0,1,2\}\hbar}.$$$$\boxed{\left(\frac{|L_z|}{|L|}\right)_{\max}=\frac2{\sqrt6}=\sqrt{2/3}=.816497}.$$

Physical check

The vector-projection sketch represents constraints on angular-momentum quantum numbers. It does not assign simultaneous definite values to all components or define a classical electron orbit.