Q003 · Spin and sequential measurement

Stern–Gerlach Sequence

Why does a middle analyzer change a repeated measurement? Because it splits the beam along another axis into two separate paths. Unless those paths are recombined with their phases intact, the coherence between them is lost, whether or not anyone reads which path each atom took. The spin selected by the first analyzer then splits again at the last one. Withdraw the middle magnet and the first result repeats.

Send a beam of neutral spin-half atoms through a sequence of magnets. Rotate the middle magnet, then keep one of its channels or let both through without reading the result. Follow the kept and blocked beams to the final analyzer, and compare the counts with a run that bypasses the middle magnet.

Model positive probability $\text{—}$

Positive / negative / rejected $\text{—}$

Current-setting trials $\text{—}$

Acquired positive fraction $\text{—}$

95% Wilson interval $\text{—}$

Output-state purity $\text{—}$

Model acceptance $\text{—}$

Physical deflection before magnification $\text{—}$

The filters are ideal projective measurements. The atoms have a positive magnetic moment and travel close to the axis. The beam splits only because the field has a gradient, and the drawing magnifies the deflection 40 times. An unread channel loses its coherence and is never recombined coherently. Input counts start after the first filter, which passes the positive channel.

No acquired record yet.

Inference from acquired counts

Acquire a scan to estimate the signal.

\text{—}

Density matrix and count conventions

\rho=\frac{I+\boldsymbol r\cdot\boldsymbol\sigma}{2}

Latest raw events · full record in CSV

Event	Setting	Outcome

Scientific basis: Feynman III.6 · MIT 8.05 · NIST Ramsey

Start with the essentials

Focus question: If nobody reads the middle result, does the spin come through unchanged?
One-sentence intuition: The middle magnet removes the coherence between its ports even when nobody reads it; sending the beam around it keeps that coherence.

Core mathematical model

Transmission between neighboring filters

P(b|a)=(1+ab\,\boldsymbol n_A\cdot\boldsymbol n_B)/2

The chance of passing the next filter depends only on the angle between the two filter axes and on which ports are selected.

An unread measurement dephases the state

\rho_{\rm unread}=\sum_{b=\pm}\Pi_b\rho\Pi_b

The part of the state for each outcome survives, but the coherence between the two outcomes is removed. Bypassing the magnet would leave the state untouched.

Deflection in a constant gradient, before display magnification

d=\frac{\mu G}{mv^2}L(D+L/2)

For atoms close to the axis, the deflection grows with the magnetic moment and the gradient and falls with the square of the speed. The screen then draws it larger.

Common difficulties

Counting only survivors hides the losses

Typical misconceptionHalf of the transmitted beam is half of the original beam.

Better mental modelWith crossed filters and the middle channel selected, only a quarter of the prepared atoms reach the final positive port. The blocked atoms must stay in the denominator.

A uniform field does not split the beam

Typical misconceptionA uniform magnetic field alone separates this neutral beam.

Better mental modelThe force comes from the gradient. A uniform field can make the spin precess, but the full beam motion in such fields lies outside this ideal projector model.

Run the experiment

01
Repeat the same axis
Choose Same axis and acquire shots.
What to observe: Every atom prepared in the positive state leaves through the positive final port.
02
Cross the middle axis
Choose Crossed filters. Predict all three count bins, then acquire shots.
What to observe: On average, half of the prepared atoms are blocked and a quarter reach each final port.
03
Leave the result unread
Choose Unread middle and acquire, then compare with Bypass middle.
What to observe: With the middle result unread, both final ports are equally likely. With the bypass, the original positive state comes back. The drawn relay keeps the two paths distinguishable.
04
Rotate the middle axis
Drag the middle axis, acquire an angle scan and check the overview, side and final collector views.
What to observe: The ports separate along the real gradient direction. A stronger gradient moves the beams farther apart, but the ideal spin overlap depends only on the angle between the axes.

Stern–Gerlach Sequence

Make a prediction, then collect the evidence.

Inference from acquired counts

What a middle analyzer changes

Core mathematical model

Transmission between neighboring filters

An unread measurement dephases the state

Deflection in a constant gradient, before display magnification

Common difficulties

Counting only survivors hides the losses

A uniform field does not split the beam

Run the experiment

Repeat the same axis

Cross the middle axis

Leave the result unread

Rotate the middle axis

Related experiments and articles

Fields, spins and precision spectra