Serial ideal Stern-Gerlach filters: a two-level model computation

Adopt P1, P2 and P3 for the declared system and assume Countable Choice, as in Declared ideal measurements and instruments of the framework. Declare a two-level system whose assigned Hilbert space is $H={\mathbb{C}}^2$ with the standard inner product and orthonormal basis $(e_{+},e_{-})$ (The state space of the framework: preparations, pure rays and density operators, Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space), interpreted as the two eigenstates of the $z$-component of a spin-half degree of freedom. Use SI units: $\mathrm{\hslash }>0$ is the adopted reduced Planck constant, with action dimension $M{L}^2{T}^{-1}$ and unit joule second ($\mathrm{J}\mathrm{s}$); the matrices $J_i$ and the angles are dimensionless. Thus $S_i$, $S_n$ and their outcome values have angular-momentum dimension $M{L}^2{T}^{-1}$ and are read in $\mathrm{J}\mathrm{s}$. No numerical value of $\mathrm{\hslash }$ is used. By the spin assignment of P6 with $s=\frac{1}{2}$, the spin operators are $S_i=\mathrm{\hslash }{J}_i$ with $J_i=\frac{1}{2}{\sigma }_i$ for the Pauli matrices in this basis (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii), Finite spin matrices clause (e)).

For a unit vector $n=(\sin \theta ,0,\cos \theta )$, $0\le \theta \le \pi$, put $u(\theta ):=\cos \frac{\theta }{2}{e}_{+}+\sin \frac{\theta }{2}{e}_{-},v(\theta ):=-\sin \frac{\theta }{2}{e}_{+}+\cos \frac{\theta }{2}{e}_{-}\mathrm{.}$ Then $u(\theta )$ and $v(\theta )$ are orthonormal and $S_n:=n\cdot \mathbf{S}=\frac{\mathrm{\hslash }}{2}(\sin \theta {\sigma }_x+\cos \theta {\sigma }_z),S_{n}u(\theta )=\frac{\mathrm{\hslash }}{2}u(\theta ),S_{n}v(\theta )=-\frac{\mathrm{\hslash }}{2}v(\theta )\mathrm{.}$ Declare the ideal filter $F(n)$ to be the ideal sharp measurement with the two outcomes $+$ and $-$, whose outcome projections are $T_{u(\theta ),u(\theta )}$ and $T_{v(\theta ),v(\theta )}$, the spectral projections of $S_n$ at $\pm \mathrm{\hslash }\mathrm{/}2$ (Declared ideal measurements and instruments of the framework clause (c), Rank-one operator calculus).

The example considers three ideal filters in series, $F_1=F(e_z)$ (so $\theta =0$), $F_2=F(n)$ with axis $n$ at angle $\theta$, and $F_3=F(m)$ with axis $m=(\sin \phi ,0,\cos \phi )$, and the initial preparation ${\rho }_0=T_{e_{+},e_{+}}$, the state of a spin-half system with $S_z=+\mathrm{\hslash }\mathrm{/}2$. Then:

(i) Through $F_1$ the outcome $+$ has probability $1$ and the conditional state is again ${\rho }_0$.

(ii) Through $F_2$, conditioned on having passed $F_1$, $\mathbb{P}(+)={\cos }^2\frac{\theta }{2},\mathbb{P}(-)={\sin }^2\frac{\theta }{2},$ and, if ${\cos }^2(\theta \mathrm{/}2)>0$, after the outcome $+$ the conditional state is the pure state of $u(\theta )$.

(iii) If ${\cos }^2(\theta \mathrm{/}2)>0$, through $F_3$, conditioned on having passed $F_2$ with the outcome $+$, the outcome $+$ has probability $\mathbb{P}(+)=\mid ⟨u(\phi ),u(\theta )⟩{\mid }^2={\cos }^2\frac{\phi -\theta }{2}\mathrm{.}$ For all $0\le \theta \le \pi$, including the zero-probability first branch, the joint probability of the outcomes $+,+,+$ is ${\cos }^2(\theta \mathrm{/}2){\cos }^2((\phi -\theta )\mathrm{/}2)$.

(iv) In particular, for $\theta =\pi \mathrm{/}2$ and $\phi =\pi$, the model probability of outcome $+$ at $F_2$ is $\frac{1}{2}$, and conditioned on that outcome the model probability of outcome $+$ at $F_3$ is also $\frac{1}{2}$, giving joint probability $\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}$ for passing all three declared filters. If $F_2$ is omitted, $F_3$ has model probability ${\cos }^2(\pi \mathrm{/}2)=0$ of outcome $+$ on the state emerging from $F_1$. Thus the intervening declared measurement changes the later model probabilities; these are not measured transmission fractions.

The computation is a model computation for declared ideal filters and the declared spin assignment: it does not derive the spatial Stern-Gerlach deflection or the inhomogeneous-field geometry, records no detector counts, resolution, background or uncertainty, and asserts nothing about the historical silver-atom experiments or about any particular apparatus.

• Idealization. The declared filters model only the spin-component outcomes and conditional updates. This example supplies no spatial field, beam-deflection or detector-response model; those are separate material and are not claimed here (Scope and empirical status of the measurement-and-instruments page).
• Order dependence is model-internal. The statement that inserting $F_2$ changes the later statistics is a consequence of the Lueders update for the declared filters; it is not a claim that every physical filter disturbs every subsequent measurement in this way, and it is not a claim about atoms in the absence of a declared preparation and detection model.
• No empirical content. The number $\frac{1}{4}$ is an exact model probability for the declared ideal filters, not a measured fraction; no calibration, efficiency, background, alignment or sampling model is supplied.