Angular Momenta and Magnetic Dipole Moments — Question 8

Question 8

A strong field reveals coincident shifts from different spin-orbital states

In a field strong enough to neglect spin-orbit coupling, a p electron has $m_l=-1,0,1$ and $m_s=\pm1/2$. Use $\delta E=\mu_BB(m_l+2m_s)$ and omit quadratic magnetic terms.

1. Find the distinct shift values and their multiplicities.

2. Calculate the mean and variance of the energy shift for equal population of the six states.

Question 8 - Solution

A strong field reveals coincident shifts from different spin-orbital states

(a) Method and calculation

$$m_s=+1/2:\ (0,1,2)\mu_BB;\qquad m_s=-1/2:\ (-2,-1,0)\mu_BB.$$$$\boxed{\delta E/\mu_BB=-2,-1,0,1,2;\quad g=1,1,2,1,1}.$$

(b) Interpretation and further calculation

$$\langle\delta E\rangle=\frac{(-2-1+0+0+1+2)\mu_BB}{6}=\boxed{0}.$$$$\boxed{\mathrm{Var}(\delta E)=\frac{4+1+0+0+1+4}{6}(\mu_BB)^2=\frac53(\mu_BB)^2}.$$

Physical check

Six basis states produce only five distinct shifts because two states coincide at zero. Multiplicity must be retained in averages.