Electron Spin — Question 8

Question 8

A coherent spin rotation transfers population sinusoidally

An electron begins in $|+z\rangle=(1,0)^T$. A supplied rotation about y is $U_y(\theta)=\begin{pmatrix}\cos(\theta/2)&-\sin(\theta/2)\\\sin(\theta/2)&\cos(\theta/2)\end{pmatrix}$.

1. Find the output spinor and its negative-z probability.

2. Compare one rotation through $\pi/3$ with two successive rotations through $\pi/3$.

Question 8 - Solution

A coherent spin rotation transfers population sinusoidally

(a) Method and calculation

$$U_y(\theta)\binom10=\binom{\cos(\theta/2)}{\sin(\theta/2)}.$$$$\boxed{P_{-z}=\sin^2(\theta/2)}.$$

(b) Interpretation and further calculation

$$\theta=\pi/3:\quad\boxed{P_{-z}=1/4}.$$$$U_y(\pi/3)^2=U_y(2\pi/3)\Rightarrow\boxed{P_{-z}=3/4}.$$

Physical check

No intermediate measurement is made. Coherent rotation angles add before probabilities are calculated; classical addition of transition probabilities would give the wrong result.