Quantum Experiments · I · Observations and models · Chapter 04
Why does inserting a different measurement change later outcomes?
In a Stern–Gerlach analyzer, an inhomogeneous magnetic field with gradient \(\partial B/\partial n\) couples a neutral particle’s magnetic dipole moment \(\boldsymbol{\mu}\) to its center-of-mass motion. For a spin-\(1/2\) beam such as ground-state silver atoms, the outgoing wave packets split into two discrete spatial channels labeled \(+n\) and \(-n\) rather than forming a continuous classical band. Blocking the lower channel with a collimator slit prepares a beam in the pure spin state \(\lvert +z\rangle\), which subsequently passes a second \(z\)-oriented analyzer with \(100\%\) transmission into the \(+z\) output.
Now insert an \(x\)-oriented Stern–Gerlach analyzer after the \(\lvert +z\rangle\) preparation stage. The \(\lvert +z\rangle\) beam splits with equal probability \(1/2\) into the \(+x\) and \(-x\) channels. If we block the \(-x\) channel so that only the \(\lvert +x\rangle\) beam continues into a final \(z\)-oriented analyzer, the \(\lvert -z\rangle\) outcome—which had been completely removed at the first stage—reappears with probability \(1/2\). A classical model that treats each measurement as merely reading out pre-existing signs \((s_x,s_z)\) without altering the state cannot reproduce this three-stage sequence.
While the historical Stern–Gerlach experiment entangles internal spin with spatial wave packets inside a magnetic field gradient, our interactive model isolates the two-dimensional spin-\(1/2\) state space and represents the two analyzer outputs as discrete port probabilities \(P(+\theta)\) and \(P(-\theta)\).
Visual animation · Section 4.1
Magnetic gradients and sequential spin analyzers
In the computational basis where \(\lvert +z\rangle=(1,0)^T\) and \(\lvert -z\rangle=(0,1)^T\), the spin-up eigenstate along an analyzer axis tilted by polar angle \(\theta\) in the \(xz\)-plane is
\[\lvert +_\theta\rangle = \cos\left(\frac{\theta}{2}\right)\lvert +z\rangle + \sin\left(\frac{\theta}{2}\right)\lvert -z\rangle, \qquad \lvert -_\theta\rangle = -\sin\left(\frac{\theta}{2}\right)\lvert +z\rangle + \cos\left(\frac{\theta}{2}\right)\lvert -z\rangle.\]
When a particle prepared in \(\lvert +z\rangle\) enters a \(\theta\)-oriented analyzer, the Born rule gives the transmission probabilities \(P(+\theta)=|\langle +_\theta\vert +z\rangle|^2=\cos^2(\theta/2)\) and \(P(-\theta)=\sin^2(\theta/2)\).
Notice the factor of \(1/2\) inside the argument \(\theta/2\): rotating the physical magnet by \(\theta=90^\circ\) (from the \(+z\) axis to the \(+x\) axis) rotates the state vector in Hilbert space by \(\theta/2=45^\circ\), yielding \(P(+x)=\cos^2(45^\circ)=1/2\). Rotating the physical magnet by \(\theta=180^\circ\) (pointing along \(-z\)) rotates the state vector by \(90^\circ\) into the orthogonal state \(\lvert -z\rangle\), giving \(P(+\theta)=\cos^2(90^\circ)=0\).
Selecting the \(+x\) channel projects the surviving particles into the state \(\lvert +x\rangle=(\lvert +z\rangle+\lvert -z\rangle)/\sqrt{2}\). When those surviving particles enter a final \(z\) analyzer, each outcome \(+z\) and \(-z\) occurs with conditional probability \(|\langle \pm z\vert +x\rangle|^2=1/2\). Relative to the original \(\lvert +z\rangle\) beam entering the middle \(x\) stage, the first selection transmits \(1/2\) of the particles and the final \(+z\) selection transmits \(1/2\) of those survivors, giving a net three-stage transmission of \((1/2)\times(1/2)=1/4\).
Visual animation · Section 4.2
Half-angle spin projections and cascaded filtering
Sequential Stern–Gerlach stages test both the Born probability rule \(|\langle \phi\vert\psi\rangle|^2\) and the Lueders–von Neumann state-update rule \(\lvert\psi\rangle\mapsto\lvert +_\theta\rangle\). An absorbing hot-wire detector that records whether an atom arrived in the \(+x\) channel without preserving a coherent \(\lvert +x\rangle\) beam has the same single-stage counting statistics as an ideal projective filter, but cannot feed a downstream \(z\) analyzer.
In laboratory atomic-beam setups, finite slit apertures, velocity dispersion, and Larmor precession in stray magnetic fields between magnets modify the ideal transmission fractions. Whenever a stage blocks one channel, experimental reports must state whether quoted probabilities are conditioned on particles surviving the intermediate slit or normalized to the initial flux leaving the source oven.
The interactive model below prepares \(\lvert +z\rangle\) and continuously varies the analyzer angle \(\theta\in[0^\circ,180^\circ]\), plotting \(P(+\theta)=\cos^2(\theta/2)\) and \(P(-\theta)=\sin^2(\theta/2)\). It demonstrates how noncommuting spin projections transform state vectors without assigning simultaneous definite values to \(S_z\) and \(S_x\).
Visual animation · Section 4.3
Projective state update and postselection accounting
Sweep the magnet angle \(\theta\) from \(0^\circ\) through \(90^\circ\) to \(180^\circ\). Verify the values of \(P(+\theta)=\cos^2(\theta/2)\) at \(\theta=60^\circ\), \(90^\circ\), and \(120^\circ\), and contrast the half-angle spin law \(\cos^2(\theta/2)\) with the full-angle optical polarizer law \(\cos^2\theta\) of Chapter 5.
Interactive model · Idealized predictions
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
A beam of \(N=800\) spin-\(1/2\) particles prepared in \(\lvert +z\rangle\) passes sequentially through an ideal \(+x\) selector and then an ideal \(+z\) selector. How many particles emerge on average after each stage, and how does this compare with removing the intermediate \(+x\) selector?
At the intermediate \(+x\) selector (\(\theta=90^\circ\)), the transmission probability is \(\cos^2(45^\circ)=1/2\), leaving an average of \(800\times(1/2)=400\) particles in the state \(\lvert +x\rangle\). At the final \(+z\) selector, the conditional transmission probability from \(\lvert +x\rangle\) into \(\lvert +z\rangle\) is again \(|\langle +z\vert +x\rangle|^2=1/2\), so an average of \(400\times(1/2)=200\) particles emerge. If the intermediate \(+x\) filter is removed, the \(\lvert +z\rangle\) beam reaches the final \(+z\) selector undisturbed and all \(800\) particles are transmitted.
What is the probability \(P(+\theta)\) when a spin-\(1/2\) particle prepared in \(\lvert +z\rangle\) enters an analyzer oriented at \(\theta=60^\circ\)?
Evaluate \(\cos^2(\theta/2)\) at \(\theta/2=30^\circ\).
Since \(\theta/2=30^\circ\) and \(\cos(30^\circ)=\sqrt{3}/2\), the probability is \(P(+60^\circ)=\cos^2(30^\circ)=3/4=0.75\).
If the intermediate stage selects the \(\lvert -x\rangle=(\lvert +z\rangle-\lvert -z\rangle)/\sqrt{2}\) channel instead of \(\lvert +x\rangle\), what is the conditional probability \(P(+z\mid -x)\) at a subsequent \(z\) analyzer?
Compute the squared magnitude \(|\langle +z\vert -x\rangle|^2\).
In the \(z\) basis, \(\lvert -x\rangle=(\lvert +z\rangle-\lvert -z\rangle)/\sqrt{2}\), so \(P(+z\mid -x)=|1/\sqrt{2}|^2=1/2\). The relative minus sign distinguishes \(\lvert -x\rangle\) from \(\lvert +x\rangle\) in an \(x\)-basis interference test, but does not change the \(z\)-basis probabilities.
Why must an experimental report specify the postselection survival fraction alongside the final outcome ratio in a multi-stage Stern–Gerlach experiment?
Compare the denominator of all prepared particles against the denominator of particles passing the intermediate slit.
Conditioning only on particles that survive intermediate slits can produce a large conditional ratio even when most prepared particles were blocked or lost. Reporting both the stage-by-stage survival fraction and the conditional outcome fraction is essential to reconstruct the full joint probabilities.
A projective spin measurement both yields random outcomes with probabilities \(\cos^2(\theta/2),\sin^2(\theta/2)\) and updates the surviving state to the selected eigenstate \(\lvert\pm_\theta\rangle\), explaining why inserting an intermediate analyzer alters downstream statistics.
These readings provide historical context, experimental implementations, or deeper derivations. The interactive display above is an educational model, not a reproduction of their measured data.