Electron Spin — Question 4

Question 4

A tilted spin eigenstate gives a half-angle probability law

For an axis in the xz plane, tilted through $\theta$ from positive z, use $|+n\rangle=\cos(\theta/2)|+z\rangle+\sin(\theta/2)|-z\rangle$. An electron is prepared in $|+z\rangle$.

1. Find the probability of a positive n measurement.

2. Find the mean n projection and evaluate both quantities at $\theta=120^\circ$.

Question 4 - Solution

A tilted spin eigenstate gives a half-angle probability law

(a) Method and calculation

$$P_{+n}=|\langle+n|+z\rangle|^2=\boxed{\cos^2(\theta/2)}.$$$$P_{-n}=1-P_{+n}=\sin^2(\theta/2).$$

(b) Interpretation and further calculation

$$\langle S_n\rangle=\frac\hbar2(P_{+n}-P_{-n})=\boxed{\frac\hbar2\cos\theta}.$$$$\theta=120^\circ:\quad\boxed{P_{+n}=1/4,\quad\langle S_n\rangle=-\hbar/4}.$$

Physical check

The probabilities sum to one, and opposite analyzer directions exchange the two outcomes. The spinor overlap uses a half angle; the expectation uses the full angle.