The Stern-Gerlach Experiment — Question 2

Question 2

A drift region amplifies the separation acquired inside the magnet

Atoms of mass M and horizontal speed v traverse a magnet of length L and gradient G, followed by a field-free drift D. Their vertical moments are $\pm\mu$, and initial vertical position and velocity are zero. Use the paraxial approximation.

1. Derive the displacement of either component at the detector.

2. Find the total separation for $\mu=9.27\times10^{-24}\,\mathrm{J/T}$, $M=1.80\times10^{-25}\,\mathrm{kg}$, $G=100\,\mathrm{T/m}$, $L=.100\,\mathrm m$, $D=.400\,\mathrm m$, and $v=500\,\mathrm{m/s}$.

Question 2 - Solution

A drift region amplifies the separation acquired inside the magnet

(a) Method and calculation

$$t_L=L/v,\quad a_z=\pm\mu G/M,\quad z_L=\tfrac12a_zt_L^2,\quad v_z=a_zt_L.$$$$\boxed{z_\pm=\pm\frac{\mu GL}{Mv^2}(D+L/2)}.$$

(b) Interpretation and further calculation

$$\Delta z=z_+-z_-=\frac{2\mu GL(D+L/2)}{Mv^2}.$$$$\boxed{\Delta z=1.854\times10^{-3}\,\mathrm m=1.854\,\mathrm{mm}}.$$

Physical check

The force acts only in the magnet, but transverse velocity persists in the drift. Ignoring the magnet-exit velocity substantially underestimates the detector separation.