The Stern-Gerlach Experiment — Question 5

Question 5

Successive perpendicular analyzers erase a previous spin selection

An unpolarized spin-half beam of flux $\Phi_0$ passes an ideal positive-z selector, then a positive-x selector, then a negative-z selector. Each selector blocks its other output. Use $|\langle\pm x|\pm z\rangle|^2=1/2$.

1. Calculate the flux after each selector.

2. Compare the final flux when the middle x selector is removed.

Question 5 - Solution

Successive perpendicular analyzers erase a previous spin selection

(a) Method and calculation

$$\Phi_{z+}=\Phi_0/2,\qquad\Phi_{x+}=\tfrac12\Phi_{z+}=\Phi_0/4.$$$$\boxed{\Phi_{z-}=\tfrac12\Phi_{x+}=\Phi_0/8}.$$

(b) Interpretation and further calculation

$$P(-z\mid+z)=|\langle-z|+z\rangle|^2=0.$$$$\boxed{\Phi_{\mathrm{without\ middle}}=0}.$$

Physical check

The middle selector prepares a new spin state. This is a sequence of selective measurements, not coherent recombination of both paths.