2026-10-25
Why the projection of spin onto a chosen axis is measured: geometry in the i-basis
In the Stern–Gerlach experiment, the apparatus selects a spin component along a particular axis. Rotate the apparatus, and it measures a component along a different direction, with the same set of possible values. Why is the measurement associated with one axis, and why can that axis be chosen arbitrarily? We will examine these questions within Wave Electricity: starting with the internal state described by the operator J, we will proceed to its spatial orientation and then to the component selected by the apparatus.
This article continues the discussion of the Stern–Gerlach experiment in the language of splitting. The central idea is to reverse the order of the geometric correspondence. When constructing a spatial representation, we specify the orientation of the internal geometry; in a measurement, the apparatus specifies the axis relative to which the state is examined. We will first clarify what it means to select a component and why the axis can be changed. We will then test the proposed description against the two outcomes for spin 1/2 and their angular probabilities.
1. A quantum example: the apparatus selects an axis
The historical Stern–Gerlach experiment used a beam of silver atoms in an inhomogeneous magnetic field. The spatial separation is associated with the magnetic moment of the atom. In the electronic ground state of silver, the electronic orbital angular momentum is zero, and the total electronic angular momentum has quantum number 1/2. Thus, in an idealized scheme that does not resolve nuclear and hyperfine structure, the two outputs are described as a measurement of a spin-1/2 projection. We will consider this ideal two-channel scheme rather than all the details of the historical apparatus.
The apparatus specifies an axis with unit vector \(\mathbf m\). Measurement of the corresponding component yields
\[\tag{1} S_{\mathbf m}=+\frac{\hbar}{2}\quad\text{or}\quad S_{\mathbf m}=-\frac{\hbar}{2}. \] Now let the first analyzer select a state with a positive projection along the unit direction \(\mathbf n\). For the selected beam, quantum mechanics predicts
\[\tag{2} P(+\mathbf m\mid+\mathbf n)=\cos^2\frac{\theta}{2},\qquad P(-\mathbf m\mid+\mathbf n)=\sin^2\frac{\theta}{2},\qquad \cos\theta=\mathbf n\cdot\mathbf m. \] When the axes coincide, the outcome of the second measurement is certain. When the axes are perpendicular, the probabilities are equal. When the axis is reversed, the previously positive state receives a negative label. Our main question comes before calculating these probabilities: why does each analyzer select a component relative to its own axis, and why does rotating the apparatus define an equally valid measurement? The unchanged set of outcomes and the angular dependence of their probabilities will serve as tests of the geometric answer.
2. The fifth postulate: one-dimensional realization
In the Wave Electricity Concept, the fifth postulate restricts each realized displacement of the center to a single local direction. On a smooth segment of the trajectory, this can be written as
\[\tag{3} d\mathbf r=\mathbf t(s)\,ds,\qquad |\mathbf t(s)|=1. \] Here, \(s\) is the arc length along the curve, and \(\mathbf t\) is its tangent. The tangent direction can change: a one-dimensional trajectory can form a circle or a spatial curve. Local one-dimensionality therefore allows a geometry with a plane and a normal.
However, the postulate concerns the displacement of the center. It does not yet determine the structure of the internal contour or imply the existence of spin 1/2. Its connection to our problem lies in distinguishing the full structure of possibilities from a single realized continuation. Observing one spin component does not, by itself, establish that the internal motion is one-dimensional.
3. From the internal state to the selected axis
We retain the notation for phase dynamics adopted in the model:
\[\tag{4} J(a,b)=\j^a(-\j)^b=\ep e^{i\pi b}+\em e^{i\pi a},\qquad \j=\ep-\em,\qquad \j^2=1. \] \[\tag{5} \ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0,\qquad \ep+\em=1. \] The parameter \(a\) describes the internal state, while \(b\) describes external motion. The idempotents select internal channels. Their phase planes are not yet coordinate planes of observable space: a spatial representation requires a separate mapping.
For the pure spin state considered here, we propose a minimal geometric correspondence. We represent the orientation of the spatial image of the internal geometry by an oriented plane with unit normal \(\mathbf n\). This normal is assigned to the prepared state \(+\mathbf n\). This is an additional correspondence with physical spin, rather than a consequence of equation (4) alone.
Orientation matters: an ordinary unoriented plane does not distinguish \(\mathbf n\) from \(-\mathbf n\). In our representation, the sign of the normal distinguishes opposite spin states. The normal characterizes the polarization of the state; it does not assign predetermined measurement outcomes along every possible axis.
In the forward construction, the internal geometry is given a spatial orientation. For measurement, we reason in the reverse order: we choose the apparatus axis and determine how the oriented state is manifested relative to it. The simplest measure of directional alignment is
\[\tag{6} \eta=\mathbf n\cdot\mathbf m=\cos\theta,\qquad -1\leq\eta\leq1. \] The vector projection of the normal onto the apparatus axis is \((\mathbf n\cdot\mathbf m)\mathbf m\), and \(\eta\) gives its signed magnitude. The chosen interaction with the analyzer selects one component. The full orientation of the state does not thereby become one-dimensional: its description contains more information than a measurement relative to a single axis. Within the adopted mapping, this interpretation accounts for the axial nature of the measurement, but it does not yet derive the apparatus interaction from the operator J.
Why can a different axis be chosen? The geometric correspondence contains no privileged spatial direction. In the absence of additional external influences, we require consistency under rotations: rotating the prepared state and the apparatus together must preserve the predictions. For an ordinary spatial rotation R, we have
\[\tag{6a} \mathbf n\mapsto R\mathbf n,\qquad \mathbf m\mapsto R\mathbf m,\qquad (R\mathbf n)\cdot(R\mathbf m)=\mathbf n\cdot\mathbf m. \] If only the apparatus is rotated, its axis becomes \(\mathbf m'\), while the preparation remains unchanged. The same rule applies to the new direction:
\[\tag{6b} \eta'=\mathbf n\cdot\mathbf m'. \] The axis is therefore not permanently attached to the internal operator: it is set by the analyzer configuration. Each new direction defines its own measurable component. This does not mean that all components can simultaneously be assigned definite numerical values for an individual particle. Changing the axis means choosing a different measurement, rather than revealing another predetermined classical projection.
The ability to use any axis is ensured here by the additional requirement that the spatial mapping be consistent under rotations. Neither the fifth postulate alone nor the two-channel expression for J proves this. Symmetry allows the probabilities to depend only on the relative angle, but it does not, by itself, determine their specific dependence on that angle.
The single number \(\eta\) cannot reconstruct the full orientation \(\mathbf n\). Reversing the order of reasoning therefore means selecting a characteristic accessible to the apparatus, rather than inverting the full spatial mapping. Next, we distinguish a continuous geometric projection from an individual quantum outcome.
4. From a continuous projection to two outcomes
An ordinary projection admits a continuous range of values. The quantity \((\hbar/2)\cos\theta\) therefore cannot be treated as the outcome of an individual spin-1/2 measurement. Instead, we adopt it as the mean over an ensemble of identically prepared systems:
\[\tag{7} \boxed{\left\langle S_{\mathbf m}\right\rangle=\frac{\hbar}{2}\,\mathbf n\cdot\mathbf m.} \] The calculation assumes two physical conditions: only the outcomes in (1) are possible, and their mean obeys (7). Geometry specifies the angular dependence of the mean, after which the probabilities are uniquely determined. This reproduces the quantum law, but it does not yet derive these two conditions from the dynamics of J.
Let \(P_+\) and \(P_-\) denote the probabilities of the two outcomes. Their sum is one, and the mean is the weighted sum of the outcomes:
\[\tag{8} P_++P_-=1,\qquad \left\langle S_{\mathbf m}\right\rangle=\frac{\hbar}{2}(P_+-P_-). \] Comparing (7) and (8) gives \(P_+-P_-=\eta\). Adding and subtracting these two linear equations yields
\[\tag{9} \boxed{P_\pm=\frac{1\pm\mathbf n\cdot\mathbf m}{2}.} \] Since \(\eta\) lies between −1 and 1, both probabilities are valid. Using the half-angle identities, we recover equation (2):
\[\tag{10} \frac{1+\cos\theta}{2}=\cos^2\frac{\theta}{2},\qquad \frac{1-\cos\theta}{2}=\sin^2\frac{\theta}{2}. \] The half-angle appears here when the probabilities are rewritten. This calculation alone does not establish the spinor periodicity of 4π or recover the phase of an amplitude; those require a more complete state structure.
The geometric projection determines the mean of the distribution. Each individual measurement produces one of the two allowed outcomes. For example, at \(\theta=60^\circ\), we obtain \(P_+=3/4\) and \(P_-=1/4\), with mean \(\hbar/4\). There is no individual outcome equal to \(\hbar/4\).
5. Sequential measurements and state preparation
To calculate the next measurement, we must describe the effect of the preceding one. For an ideal measurement with selection of a particular output, we adopt the update rule
\[\tag{11} \mathbf n'=\sigma\mathbf m,\qquad \sigma=+1\, \text{or}\ -1. \] Here, \(\sigma\) labels the recorded output. After a positive outcome, the state is prepared along \(\mathbf m\); after a negative outcome, it is prepared along \(-\mathbf m\). If the measurement is immediately repeated along the same axis, without intervening evolution, equation (9) gives the same outcome with probability one.
For the next analyzer, with unit axis vector \(\mathbf l\) and outcome \(\tau=\pm1\), we have
\[\tag{12} P(\tau\mathbf l\mid\sigma\mathbf m)=\frac{1+\tau\sigma\,\mathbf m\cdot\mathbf l}{2}. \] Consider the sequence: preparation in \(+z\), measurement along \(x\), and then measurement along \(z\). After selecting the \(+x\) output, the final measurement gives \(+z\) and \(-z\), each with probability 1/2. If the intermediate ideal measurement along \(x\) is recorded but its outcome is subsequently forgotten, the resulting state is a mixture of \(+x\) and \(-x\); the final measurement along \(z\) again gives equal probabilities. In both cases, the intermediate interaction has changed the preparation.
A mere rotation of the mathematical notation produces no such change. Nor should a completed measurement be identified with coherent beam splitting followed by reversible recombination: in the latter case, relative phases may be preserved. Rule (11) applies to the state after an ideal measurement or selection, rather than to every operation on the channels.
6. Connections with the description of quantum phenomena
The article on the wave function distinguishes channel weights from their phases. For our problem, this means that an unchanged diagonal J does not yet specify how probabilities are redistributed as the angle changes. Its connection to the analyzer is introduced through orientation and the mean-value rule. The article on quantum interpretations distinguishes the full state from an individual history of measurement outcomes. Here, that distinction allows us to discuss a single outcome without prematurely replacing the full state by a randomly selected channel.
The diffraction and interference problems considered in the same section require relative phases and the addition of amplitudes. The probability formula derived here therefore does not replace the general wave formalism. Its scope is ideal measurements of a single spin degree of freedom with a specified preparation and no unaccounted evolution between analyzers. Mixed preparations can be treated by averaging over initial orientations. Joint states of several particles require a separate description.
7. Can interchanging the parameters of J select an outcome?
Consider the hypothesis under discussion: two internal phase orientations are related by the interchange \(J(a,b)\leftrightarrow J(b,a)\). Introduce the common and relative phases:
\[\tag{13} \Phi=\frac{\pi(a+b)}{2},\qquad \delta=\frac{\pi(a-b)}{2}. \] Direct substitution into (4) gives
\[\tag{14} \begin{aligned} J(a,b)&=e^{i\Phi}\bigl(\cos\delta-i\j\sin\delta\bigr),\\ J(b,a)&=e^{i\Phi}\bigl(\cos\delta+i\j\sin\delta\bigr). \end{aligned} \] The common phase remains unchanged, while the relative phase changes sign. This does not yet prove that the two alternatives correspond to opposite spins: at \(a=b\), they coincide, and in general their action need not produce orthogonal states. Moreover, interchanging parameters that describe the internal state and external motion requires a physical justification.
If these alternatives can eventually be associated with the apparatus outputs, the random choice must account for the orientations \(\mathbf n\) and \(\mathbf m\) and have the weights in (9). An independent choice with probabilities 1/2 would fail even to reproduce repeatability along the same axis. The interchange therefore remains a candidate for an internal mechanism, rather than an established explanation of measurement.
8. Why the axis can be changed
In the proposed description, the internal geometry is given a spatial orientation, and the analyzer selects a characteristic relative to its own axis. Consistency under rotations allows the same rule to be applied to any apparatus direction. The axial nature of the measurement is therefore compatible with a free choice of axis: the correspondence rule is fixed, while the particular setup specifies the direction.
The two allowed outcomes and the adopted mean-value law yield the probabilities \(\cos^2(\theta/2)\) and \(\sin^2(\theta/2)\); the state-update rule describes subsequent ideal measurements. These checks show that the scheme is consistent with the spin statistics considered here. The next step is to derive the spatial mapping itself and the interaction with the analyzer from the dynamics of the model.
The Wave Electricity Concept specifies the local one-dimensionality of realized motion, while the phase channels retain the description of possibilities. Spatial orientation and the apparatus axis connect these levels in the problem considered here. The next substantive step is to construct a physical interaction with the analyzer that justifies the mean-value law, the two outputs, and the state update within the model itself. The convenient geometric correspondence could then become a dynamical explanation.