Discussion
Two Stern-Gerlac apparatus \(S_{1}\) and \(S_{2}\) are kept along the z-axis with the directions of their magnetic fields along the positive x and y directions respectively. Each apparatus allows only those particles to pass which have their spins aligned in the direction of its magnetic field. If an unpolarised beam of spin ½ particles is made incident on \(S_{1}\), the ratio of the intensities \(I_{i} : I_{f}\) of the initial and final beams is :
Answer: B
The ratio \(I_{i} : I_{f}\) is 4 : 1.
An unpolarised beam of spin-1/2 particles splits equally along any axis, so \(S_{1}\), passing only spin +x, transmits half the beam. Those particles are in the state \(|+x\rangle\). In the y basis
\[ |\langle +y|+x\rangle|^{2} = \frac{1}{2} \]
so \(S_{2}\) passes half of what reaches it. The final intensity is \(\frac{1}{2}\times\frac{1}{2} = \frac{1}{4}\) of the initial, a ratio of 4 : 1.
2 : 1 counts only the first apparatus. 1 : 0 would follow if x and y spin states were orthogonal, but only opposite directions, such as +y and -y, are orthogonal for spin 1/2. 8 : 1 would need a third filter.
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