2026-10-18
Why does a repeated measurement change the result?
The Stern–Gerlach experiment and splitting
A beam of atoms passes through a device that splits it into two beams. One beam is blocked, while the other is sent into an identical device. With the same settings, the second device no longer splits the beam: all the atoms leave through a single output. But if a third device, rotated relative to the others, is placed between them and only one of its beams is allowed through, the final device splits the atoms into two equal groups again. Why does the intermediate measurement change the result of the next one? We will examine the experiment step by step and then show how to describe it by splitting the state of a particle into channels.
1. What a Stern–Gerlach Analyzer Measures
In a nonuniform magnetic field, a particle beam can split according to the component of its magnetic moment along a chosen direction. For an idealized spin-1/2 system, the analyzer has two outputs. We will label them ‘+’ and ‘−’. In this article, ‘up’ and ‘down’ refer to the signs of the spin component along the analyzer axis, not to a small ball rotating about its own axis.
The original Stern–Gerlach experiment used silver atoms. Here we consider an instructional arrangement of successive ideal spin analyzers, similar to the one shown in the video. The mathematical description applies to a two-level spin state. An implementation with free electrons would also need to account for the Lorentz force and the design of the apparatus.
The z-axis defines the first measurement direction. The x-axis is perpendicular to it. The z and x analyzers therefore ask different questions about the same state: what is the spin component along z, and what is the spin component along x?
For spin 1/2, the possible component values are \(+\hbar/2\) and \(-\hbar/2\). Throughout the following discussion, we assume ideal analyzers: there is no additional spin rotation between them, and passage through a selected output does not itself change the selected spin state. The positions of the branches in the figure are schematic: the output signs label spin components, whereas deflection in an actual device depends on the magnetic moment and the field gradient.
2. Three Successive Measurements
First, we send the particles through a z analyzer and block the −z output. The beam emerging from the first filter is then in the +z state. We will take its intensity as 100%. If another z analyzer is placed immediately after it, this entire beam leaves through +z. Under ideal conditions, the −z output remains empty.
Now place an x analyzer between the two z measurements. It splits the prepared +z beam into two equal parts: +x and −x. Block −x and retain +x. Half of the prepared beam reaches the final analyzer. This last z analyzer splits it equally again, into +z and −z. Each final output contains 25% of the beam after the first filter, or 50% of the beam that reaches the last measurement.
Fig. 1 Intermediate selection along x changes the outcome of the next measurement along z. SG denotes a Stern–Gerlach analyzer. Crosses mark blocked outputs.
At first, it may seem that the second device has erased an earlier property of the particle. More precisely, selecting +x prepares a different spin state. The record of the first +z result has not disappeared. It simply no longer guarantees the outcome of a later z measurement.
3. One State and Two Channels
In the language of splitting, a state has two components relative to the chosen measurement [1]. For the z-axis, introduce normalized, orthogonal channel states:
\[\tag{1} \langle+z|+z\rangle=\langle-z|-z\rangle=1,\qquad\langle+z|-z\rangle=0. \] The notation \(|+z\rangle\) denotes a state with a positive spin component along z; \(|-z\rangle\) denotes one with a negative component. The relation \(\langle+z|-z\rangle=0\) shows that these states are orthogonal. A pure state can be described by a single vector \(\Psi\), unlike a statistical mixture of several prepared states. Its general form in the chosen basis is:
\[\tag{2} \Psi=c_+|+z\rangle+c_-|-z\rangle,\qquad |c_+|^2+|c_-|^2=1. \] The numbers c₊ and c₋ are amplitudes. Under the adopted Born rule, the probabilities of the outputs are the squared magnitudes of these amplitudes. They describe the possible measurement outcomes for a single state. This notation does not imply the existence of two separate particles.
Introduce idempotents that select the corresponding components:
\[\tag{3} \ep_z=|+z\rangle\langle+z|,\qquad\em_z=|-z\rangle\langle-z|. \] \[\tag{4} \ep_z^2=\ep_z,\qquad\em_z^2=\em_z,\qquad\ep_z\em_z=0,\qquad\ep_z+\em_z=I. \] Here and below, idempotents are represented as operators on the state space. I denotes the identity operator. Selecting an already selected branch again leaves it unchanged. Selecting the opposite branch from that same state gives zero. This is why repeated measurement along the same axis preserves the selected outcome.
4. What the Operator J Adds
As in the original hypothesis, the phase dynamics of two fixed channels can be written using the operator:
\[\tag{5} J(a,b)=\j^a(-\j)^b=\ep e^{i\pi b}+\em e^{i\pi a}. \] The parameter a retains its meaning as the internal state, while b describes external motion. In a representation where these idempotents select two orthogonal channels, J multiplies their amplitudes by phase factors. The squared magnitudes of the amplitudes are preserved in this fixed basis. Measurement probabilities in another basis may depend on the relative phase.
A diagonal J alone therefore does not explain changes in the weights of the original outputs. To describe the experiment, we must also specify how different measurement directions are related. The spin projectors indexed by z and x represent these directional decompositions. Their physical correspondence with the internal channels of the model is established through a separate spin mapping; the directional indices cannot be assigned to the internal channels without this step.
5. Deeper Splitting and a Change of Direction
Nested splitting reveals further possibilities within an already selected branch. Their probability weights add up to the weight of the parent branch. This is useful for describing a sequence of events, but it does not by itself specify a rotation of the spin analyzer.
Rotating the analyzer defines another decomposition of the same state space. A branch that belongs entirely to one z output may have two nonzero components relative to the x outputs. Describing this requires a relation between bases, rather than merely adding levels to a fixed tree of channels.
A change in mathematical notation alone makes no physical difference. The same state can be expressed in a new basis and then converted back. The prepared state changes through physical interaction and selection of a particular outcome.
6. Why the Half-Angle Appears
The article on electron spin considers a wave that returns to its initial state after successively traversing both sheets of an internal contour. The full path corresponds to an angle of traversal of \(4\pi\), while the phase of the fundamental mode changes by \(2\pi\) over that path. Measured from a common origin, the wave phase \(\chi\) is therefore half the internal angle of traversal \(\theta_{\mathrm{int}}\):
\[\tag{6} \chi=\frac{\theta_{\mathrm{int}}}{2}. \] This relation is obtained in the approximation that neglects the lengths of the transitions between the sheets. It concerns the internal traversal of the wave. To describe rotation of an analyzer, the cited article separately adopts a spin representation of physical rotations, which also involves a half-angle. We will use that representation while distinguishing the internal angle of traversal from the angle between the axes of the instruments.
For an axis rotated through an angle θ in the xz-plane, the new states can be chosen as follows:
\[\tag{7} \begin{aligned}|+\theta\rangle&=\cos\frac\theta2\,|+z\rangle+\sin\frac\theta2\,|-z\rangle\-\theta\rangle&=-\sin\frac\theta2\,|+z\rangle+\cos\frac\theta2\,|-z\rangle.\end{aligned} \] In the two-dimensional state space, these formulas are expressed by a rotation matrix. Its columns contain the components of the new basis states relative to the old ones:
\[\tag{8} U(\theta)=\begin{pmatrix}\cos(\theta/2)&-\sin(\theta/2\sin(\theta/2)&\cos(\theta/2)\end{pmatrix},\qquad U^\dagger U=I. \] The symbol \(\dagger\) denotes the Hermitian adjoint: transposition followed by complex conjugation. Here the matrix is real, so transposition alone is sufficient. Acting with \(U\) on a state describes an active spin rotation. When we merely express the same state in a new basis, its coefficients transform with the inverse matrix \(U^\dagger\), while the state itself remains unchanged.
These are two orthogonal directions in spin state space. The half-angle appears in their amplitudes. Applying the Born rule to the prepared +z state gives:
\[\tag{9} \boxed{P(+\theta\mid+z)=\cos^2\frac\theta2,\qquad P(-\theta\mid+z)=\sin^2\frac\theta2.} \] When the axes coincide, the first outcome has probability 1. When they are perpendicular, both probabilities are 1/2. Reversing the axis assigns the opposite output label to the same spin state.
7. Splitting, Beam Separation, and Selection
For each analyzer orientation, define a pair of projectors:
\[\tag{10} \ep_\theta=|+\theta\rangle\langle+\theta|,\qquad\em_\theta=|-\theta\rangle\langle-\theta|. \] Their relation to the original splitting direction now takes an explicit form:
\[\tag{11} \ep_\theta=U(\theta)\ep_zU^\dagger(\theta),\qquad\em_\theta=U(\theta)\em_zU^\dagger(\theta). \] The new pair of idempotents is thus obtained by rotating the original decomposition. Channel orthogonality and the projector sum I are preserved. This action on state space supplements the internal commutative algebra of the model.
The ideal splitter correlates the two spin components with different spatial paths. Denote the amplitudes in the basis of this analyzer by \(d_+=\langle+\theta|\Psi\rangle\) and \(d_-=\langle-\theta|\Psi\rangle\). In general, they differ from the coefficients \(c_+,c_-\) in the z basis. If \(\Phi_0\) is the incoming spatial wave packet and \(\Phi_+,\Phi_-\) are the outgoing packets, the process is schematically:
\[\tag{12} (d_+|+\theta\rangle+d_-|-\theta\rangle)\Phi_0\longrightarrow d_+|+\theta\rangle\Phi_++d_-|-\theta\rangle\Phi_-. \] After separation, both branches can still be considered together. When one output is blocked, we consider only the particles that pass through the other. Such selection gives the following conditional state update:
\[\tag{13} P(+\theta)=\|\ep_\theta\Psi\|^2,\qquad\Psi_{+\theta}=\frac{\ep_\theta\Psi}{\sqrt{P(+\theta)}},\qquad P(+\theta)>0. \] The denominator restores unit normalization within the group of transmitted particles. It does not restore the lost intensity: part of the original beam remains excluded. The formula specifies the conditional state after successful selection; it does not explain the mechanism of an individual detection event.
8. Returning to the Three Analyzers
After the first filter, the state is |+z⟩. For the perpendicular analyzer, θ = π/2. Therefore:
\[\tag{14} |+x\rangle=\frac{|+z\rangle+|-z\rangle}{\sqrt2},\qquad|-x\rangle=\frac{-|+z\rangle+|-z\rangle}{\sqrt2}. \] The initial state has two amplitudes of equal magnitude in the new decomposition:
\[\tag{15} |+z\rangle=\frac{|+x\rangle-|-x\rangle}{\sqrt2}. \] The +x filter transmits half the beam and prepares the state |+x⟩. Relative to the third analyzer, this state contains both z components:
\[\tag{16} \ep_z|+x\rangle=\frac{|+z\rangle}{\sqrt2},\qquad\em_z|+x\rangle=\frac{|-z\rangle}{\sqrt2}. \] The two outcomes are therefore equally likely among the particles reaching the third analyzer. If N particles pass the first +z selection, then on average N/2 pass the +x filter, and each final output receives N/4 particles. Splitting thus gives the numbers shown in the figure.
In operator notation, the reappearance of the previously excluded output is expressed by:
\[\tag{17} \em_z\ep_x\ep_z=\frac12|-z\rangle\langle+z|\ne0. \] For mutually commuting projectors, this product would be zero: the factors could be reordered to obtain the product of opposite z projectors. Directional spin splittings therefore require operators that do not generally commute. They act on state space; this structure supplements the commutative algebra of the internal phase channels.
If the middle analyzer is rotated through an arbitrary angle θ and only its positive branch is still transmitted, we can predict the fraction reaching the final −z output relative to the beam after the first filter:
\[\tag{18} \frac{I_{-z}^{\mathrm{final}}}{I_{+z}^{\mathrm{prepared}}}=\cos^2\frac\theta2\,\sin^2\frac\theta2=\frac14\sin^2\theta. \] The first factor gives the fraction of particles transmitted by the intermediate filter; the second gives the fraction of −z outcomes among those transmitted. For perpendicular axes, the result is 1/4. When the axes coincide, the negative output is absent. This allows the explanation to be tested against the full angular dependence, rather than only a single 50% value.
9. What If Neither Intermediate Outcome Is Selected?
Consider a different arrangement: neither x branch is excluded, but information about the output is retained in the apparatus or another physical record and then ignored when the results are counted. The particles must remain available for the next measurement; for example, the intermediate detection must be nondestructive. Equivalent statistics can be obtained by analyzing both spatial branches separately along z and combining the results. Ideally, the spin part of the state is described by a mixture:
\[\tag{19} \rho=\frac12|+x\rangle\langle+x|+\frac12|-x\rangle\langle-x|=\frac I2. \] The matrix ρ describes spin measurement statistics when the intermediate outcomes are not distinguished. The two terms represent equal weights of the +x and −x states. For any subsequent spin analyzer, this mixture gives probabilities of 1/2 and 1/2. If there are no losses and both branches are included, each final z outcome accounts for 50% of the prepared +z beam. This differs from the arrangement in the figure, where the filter excludes half the beam.
The same matrix describes the spin part alone when the two orthogonal spatial paths remain separated and are not taken into account. The full state of spin and paths may still retain coherence. If the separation is precisely reversed, restoring the spatial packets and relative phases and removing all path-distinguishing markers, the original +z state can be restored. If a path record remains in the apparatus or environment, merely recombining the beams is insufficient.
We therefore need to know exactly what happens to the branches: whether one is selected, both outcomes are recorded and ignored, or the separation is coherently reversed. Saying that a second analyzer was inserted does not fully specify the experiment without these details.
10. What the Experiment Tells Us About Splitting
The first selection prepares a state with a definite spin component along z. Intermediate selection along x prepares a different state. That state again has two components along z. The final device therefore has an output that was absent when the first measurement was repeated directly. An earlier recorded result remains valid for the earlier stage of the experiment.
A state that belongs entirely to one branch of a chosen splitting may have two nonzero components relative to another direction. Successive branch selection changes the prepared state and hence the probabilities of subsequent outcomes.
In this construction, idempotents select the channels, the spin representation relates analyzer directions, and the Born rule converts amplitudes into probabilities. This gives a consistent description of the experiment in the language of splitting. An independent geometric derivation must also justify the mapping from the internal two-sheet structure to physical spin rotations, interaction with the apparatus, and the detection rule. The internal sheets are not identified with the two spatial outputs of the analyzer.
The idea of a more general state, of which observable reality might be one channel, belongs to a separate extension of the original hypothesis. It is not required to calculate the results of this experiment.
References
- The Wave Function as a Representation of a Split Event — the original version of the hypothesis.
- Electron Spin as a Consequence of the Two-Sheet Cycle of a Closed Wave.
- The Feynman Lectures on Physics. Vol. III, Chapter 6: Spin One-Half.
- How Quantum Measurement Erases the Past — an illustration of successive spin measurements. [YouTube]