Electron Spin — Question 3

Question 3

A relative phase becomes measurable after changing the spin basis

Let $|\chi\rangle=(|+z\rangle+e^{i\phi}|-z\rangle)/\sqrt2$. The x eigenstates are $|\pm x\rangle=(|+z\rangle\pm|-z\rangle)/\sqrt2$.

1. Derive both x probabilities as functions of phase.

2. Evaluate them for $\phi=2\pi/3$ and compare with the z probabilities.

Question 3 - Solution

A relative phase becomes measurable after changing the spin basis

(a) Method and calculation

$$\langle\pm x|\chi\rangle=(1\pm e^{i\phi})/2.$$$$\boxed{P_{\pm x}=(1\pm\cos\phi)/2}.$$

(b) Interpretation and further calculation

$$\cos(2\pi/3)=-1/2\Rightarrow\boxed{P_{+x}=1/4,\quad P_{-x}=3/4}.$$$$\boxed{P_{+z}=P_{-z}=1/2}\quad\text{for every }\phi.$$

Physical check

A common phase multiplying the entire spinor is unobservable, but a relative phase changes interference in another measurement basis.