The Stern-Gerlach Experiment — Question 7

Question 7

Discarding the analyzer record differs from undoing its interaction

Initially the spin is $|+x\rangle=(|+z\rangle+|-z\rangle)/\sqrt2$. A z analyzer correlates the states with orthogonal paths A and B. In case I the path information is irreversibly discarded; in case II the analyzer interaction is exactly reversed before detection.

1. Find the reduced spin state in case I and the positive-x probability.

2. Find the positive-x probability in case II.

Question 7 - Solution

Discarding the analyzer record differs from undoing its interaction

(a) Method and calculation

$$|\Psi\rangle=(|+z,A\rangle+|-z,B\rangle)/\sqrt2,\quad\langle A|B\rangle=0.$$$$\rho_{\mathrm{spin}}=\mathrm{Tr}_{\mathrm{path}}|\Psi\rangle\langle\Psi|=I/2\Rightarrow\boxed{P_I(+x)=1/2}.$$

(b) Interpretation and further calculation

$$U^{-1}|\Psi\rangle=|+x\rangle|\text{input path}\rangle.$$$$\boxed{P_{II}(+x)=1}.$$

Physical check

Exact reversal includes phase control and erasure of every path record. Merely bringing two incoherent beams to the same location does not restore their spin coherence.