The Stern-Gerlach Experiment — Question 9

Question 9

Gravity shifts the midpoint but leaves magnetic splitting unchanged

Two neutral spin components have accelerations $a_z=\pm a_m-g$ throughout a magnet transit t, followed by drift time $t_d$ during which only gravity acts. Initially z and vertical velocity vanish.

1. Derive both detector positions.

2. Find their midpoint and separation and identify how to extract g and $a_m$.

Question 9 - Solution

Gravity shifts the midpoint but leaves magnetic splitting unchanged

(a) Method and calculation

$$z_\pm=\tfrac12(\pm a_m-g)t^2+(\pm a_m-g)tt_d-\tfrac12gt_d^2.$$$$\boxed{z_\pm=\pm a_mt(t_d+t/2)-\tfrac12g(t+t_d)^2}.$$

(b) Interpretation and further calculation

$$\boxed{\bar z=(z_++z_-)/2=-g(t+t_d)^2/2}.$$$$\boxed{\Delta z=2a_mt(t_d+t/2)};\quad g=-2\bar z/(t+t_d)^2.$$

Physical check

Gravity is common to both branches and cancels in their difference. The midpoint is measured relative to the undeflected straight trajectory.