= Stern-Gerlach expectation estimation
{c}
A concrete idealized implementation uses a <Stern-Gerlach experiment> twice: a preparation filter along $\mathbf n=(\sin\theta,0,\cos\theta)$ passes only its plus <spin> channel, and a $z$ analyzer resolves both output channels for many fresh atoms. This prepares the <spin one-half along an axis> state $|\psi_\theta\rangle$. The authored apparatus description uses slit collimation, a nonuniform <magnetic field> with a pointed pole, and counters at the separated outputs.<The Feynman Lectures on Physics, Volume III by Richard Feynman and others/Chapter 5: Spin One>[The Feynman Lectures on Physics, Volume III, Chapter 5: Spin One, §5–1, figure 5–1 and the paragraph proposing output counters; §5–4, slit diffraction and internal-state limitations]{ref} The two-channel <spin one-half> preparation and analyzer are described separately; the source's recombining “improved” apparatus is explicitly idealized, so the setup here is a proposed estimation model, not a report of observed counts.<The Feynman Lectures on Physics, Volume III by Richard Feynman and others/Chapter 6: Spin One-Half>[The Feynman Lectures on Physics, Volume III, Chapter 6: Spin One-Half, §6–2, figures 6–1 and 6–2: two beams, one-channel preparation, and a second analyzer; idealized apparatus qualification in §5–1 of the preceding chapter]{ref}
Count every accepted analyzer event in its calibrated $+z$ or $-z$ channel and compute $\widehat\mu_N$ from the count difference. The <magnetic field> gradient separates trajectories through their <magnetic moments>; calibrated channel labels convert positions into <spin> <eigenvalues>. The readout is a population imbalance, not an intermediate trajectory carrying <spin> $\hbar\cos\theta/2$. Beam position also depends on flight time and field geometry, so averaging raw positions alone does not establish a calibrated <spin expectation value>.
As controls, feed known $|+z\rangle$ and $|-z\rangle$ preparations to check the channel assignment and leakage, hold the analyzer axis fixed, and repeat short blocks to check preparation drift. Measure background counts and relative channel efficiencies. With efficiencies $\eta_+$ and $\eta_-$ and negligible background, the detected plus fraction is
$$
p_+^{\rm det}=\frac{\eta_+p_+}{\eta_+p_++\eta_-(1-p_+)}.
$$
Unequal efficiencies bias the naive <sample mean>, even for arbitrarily many counts. For example, $p_+=3/4$, $\eta_+=1$, and $\eta_-=1/2$ give detected fraction $6/7$ and naive mean $5\hbar/14$, rather than $\hbar/4$. Background, overlapping beams, detector saturation, angle misalignment, unwanted internal states, and preparation correlations require a better <quantum measurement> model. Increasing $N$ only reduces the independent sampling contribution; it cannot remove these systematic errors. The selected preparation beam is the population being estimated, so losses in preparing it must be distinguished from losses in reading its two outcomes.
Back to article page
