The Stern-Gerlach Experiment — Question 8

Question 8

More than two spots reveal a larger total angular momentum

Neutral atoms have $J=3/2$ and magnetic projections $\mu_z=-g_J\mu_Bm_J$, with $g_J=4/3$. A gradient G acts for time t with no drift. Initial transverse velocities vanish and all four sublevels are equally populated.

1. Find the detector positions after the magnet in terms of $K=\mu_BGt^2/(2M)$.

2. Find the adjacent spacing and intensity fraction of each spot.

Question 8 - Solution

More than two spots reveal a larger total angular momentum

(a) Method and calculation

$$z=\tfrac12(\mu_zG/M)t^2=-g_Jm_JK,\quad m_J=-3/2,-1/2,1/2,3/2.$$$$\boxed{z/K=2,\ 2/3,\ -2/3,\ -2}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta z_{\mathrm{adj}}=4K/3}.$$$$\boxed{\Phi_{\mathrm{each}}/\Phi_0=1/(2J+1)=1/4}.$$

Physical check

Number of spots measures available projections when all are populated and resolved. Four spots do not mean four possible electron spin projections; J includes other angular momentum.