The Stern-Gerlach Experiment — Question 1

Question 1

A Stern-Gerlach force depends on the field gradient

A neutral atom has a fixed adiabatic magnetic-moment projection $\mu_z=\pm\mu$. Across the beam in the vertical direction the field magnitude is $B(z)=B_0+Gz$, with $G>0$. Use $U=-\mu_zB$ and neglect spin flips.

1. Derive the vertical force for each component.

2. Compare increasing $B_0$ at fixed G with increasing G at fixed $B_0$.

Question 1 - Solution

A Stern-Gerlach force depends on the field gradient

(a) Method and calculation

$$F_z=-\partial U/\partial z=\mu_z\,dB/dz.$$$$\boxed{F_{z,\pm}=\pm\mu G}.$$

(b) Interpretation and further calculation

$$\left.\frac{\partial F_z}{\partial B_0}\right|_G=\boxed{0},\quad\left.\frac{\partial F_z}{\partial G}\right|_{B_0}=\mu_z.$$$$G\to2G\Rightarrow\boxed{F_{z,\pm}\to2F_{z,\pm}}.$$

Physical check

A uniform bias field can define the quantization direction without separating the beam. The model describes the local field magnitude; a real magnet also has transverse field components consistent with Maxwell equations.