Magnetism and Electrons — Question 2

Question 2

The orbital quantum number fixes a moment magnitude but not a unique projection

An electron has orbital quantum number l=2. Use $|\mathbf L|=\hbar\sqrt{l(l+1)}$, $L_z=m_l\hbar$, $m_l=-l,\ldots,l$, and $\mu_B=e\hbar/(2m_e)$. Ignore spin.

1. Find the orbital moment magnitude and all allowed z projections.

2. For $m_l=2$ find the projection-to-magnitude ratio and explain why the largest projection is smaller than the magnitude.

Question 2 - Solution

The orbital quantum number fixes a moment magnitude but not a unique projection

(a) Method and calculation

$$|\boldsymbol\mu_L|=\frac{\mu_B}{\hbar}|\mathbf L|=\boxed{\sqrt6\mu_B}.$$$$\boxed{\mu_{Lz}=-m_l\mu_B\in\{2,1,0,-1,-2\}\mu_B}.$$

(b) Interpretation and further calculation

$$\left.\frac{\mu_{Lz}}{|\boldsymbol\mu_L|}\right|_{m_l=2}=-\frac{2}{\sqrt6}.$$$$\boxed{\max|\mu_{Lz}|=2\mu_B<\sqrt6\mu_B}.$$

Physical check

Quantum magnitude and a measured component are different observables; the projection diagram is not a set of classical tilted orbits.