The Stern-Gerlach Experiment — Question 6

Question 6

A tilted intermediate selector maximizes transmission between opposite filters

A beam already prepared in $|+z\rangle$ passes a positive-n selector followed by a negative-z selector. Axis n is tilted through $0\le\theta\le\pi$ from z. Use the spin-half overlap rule $P(+n|+z)=\cos^2(\theta/2)$.

1. Find the transmitted fraction as a function of angle.

2. Find the angle and maximum fraction.

Question 6 - Solution

A tilted intermediate selector maximizes transmission between opposite filters

(a) Method and calculation

$$P(-z|+n)=\sin^2(\theta/2),\quad T=\cos^2(\theta/2)\sin^2(\theta/2).$$$$\boxed{T(\theta)=\tfrac14\sin^2\theta}.$$

(b) Interpretation and further calculation

$$\frac{dT}{d\theta}=\tfrac12\sin\theta\cos\theta=0.$$$$\boxed{\theta_{\max}=\pi/2,\quad T_{\max}=1/4}.$$

Physical check

This fraction refers to the already prepared input. An additional initial positive-z selection from an unpolarized source would halve the total source-to-detector transmission.