The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4, the kinematics and massive-particle model postulate P5 and the internal-factor assignment of the spin postulate P6, read for the massive spin-$\frac{1}{2}$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m>0$ and spin label $s=\frac{1}{2}$, together with the effective magnetic-moment coupling of Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator with parameter ${\gamma }_{1\mathrm{/}2}\in \mathbb{R}$ and the angular momentum readings of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations), and assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $B_z\in C^1(\mathbb{R})$ be real with $B_z$ and $B_z^{\mathrm{\prime }}$ bounded, let $\mathbf{B}(x):=(0,0,B_z(x_1)),$ let $H_Z$ be the Zeeman operator of the magnetic-moment postulate for this profile, acting on $H_{1\mathrm{/}2}=L^2({\mathbb{R}}^3;{\mathbb{C}}^2)$ with domain $D(H_Z)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^2)$, and let $u(t)=e^{-it{H}_Z\mathrm{/}\mathrm{\hslash }}{u}_0$ be its flow. Identify $H_{1\mathrm{/}2}$ with $\mathcal{H}=L^2({\mathbb{R}}^3;\mathbb{C})\oplus L^2({\mathbb{R}}^3;\mathbb{C})$ through the internal basis $(e_{1\mathrm{/}2},e_{-1\mathrm{/}2})$, and let $P_{+}:=\frac{1}{2}(I+2S_3\mathrm{/}\mathrm{\hslash })$ be the fibrewise orthogonal projection onto the spin-up line $e_{1\mathrm{/}2}$. Then:

(a) under this identification $H_Z$ is the operator $K(u_{+},u_{-})=(\frac{1}{2m}{\sum }_j{P}_j^2{u}_{+}-\frac{{\gamma }_{1\mathrm{/}2}\mathrm{\hslash }}{2}{B}_z(Q_x)u_{+},\frac{1}{2m}{\sum }_j{P}_j^2{u}_{-}+\frac{{\gamma }_{1\mathrm{/}2}\mathrm{\hslash }}{2}{B}_z(Q_x)u_{-}),D(K)=W^{2,2}\oplus W^{2,2},$ of Conditional Stern-Gerlach force for a spin-coupled bounded field model with ${\gamma }_s:={\gamma }_{1\mathrm{/}2}$; consequently $H_Z$ is self-adjoint, $S_3{H}_Z=H_Z{S}_3$ on $D(H_Z)$, the sector weights $\mathrm{\parallel }{P}_{\pm }u(t){\mathrm{\parallel }}^2$ and the mean $⟨S_3{⟩}_{u(t)}$ are independent of $t$;

(b) if in addition $u_0\in D(H_Z^2)$ is a unit vector with $P_{+}{u}_0=u_0$ and $B_z^{\mathrm{\prime }}(x_1)=G$ for every $x_1$ in a Borel set $N\subseteq \mathbb{R}$ satisfying ${\int }_{\mathbb{R}\setminus N}{\eta }_{+}(x_1)d{x}_1=0$, where ${\eta }_{+}(x_1):={\int }_{{\mathbb{R}}^2}\mathrm{\mid }{u}_0^{+}(x_1,x_2,x_3){\mathrm{\mid }}^{2}d{x}_{2}d{x}_3$ is the first-coordinate marginal density of the spin-up component, then $\frac{d}{dt}{\mid }_{t=0}⟨P_{x}u(t),u(t)⟩=\frac{{\gamma }_{1\mathrm{/}2}\mathrm{\hslash }}{2}G;$ for $t\mathrm{\ne }0$ the general rate is $\frac{d}{dt}⟨P_{x}u(t),u(t)⟩={\gamma }_{1\mathrm{/}2}⟨S_3{B}_z^{\mathrm{\prime }}(Q_x)u(t),u(t)⟩$;

(c) the profile satisfies $\mathrm{div}\mathbf{B}=0$ and $\mathrm{curl}\mathbf{B}=(0,-B_z^{\mathrm{\prime }}(x_1),0)$ at every point of ${\mathbb{R}}^3$, as identities of the displayed components;

(d) Scope. The statement is conditional on P1, P2, P4, P5, P6 and the adopted coupling. Clause (b) is a model prediction for the rate of the mean transverse momentum of a prepared spin-up state under the marginal-support hypothesis; it is not an observation of a beam splitting, a trajectory or a count, it identifies no apparatus, calibration or detector, and it asserts neither a source-free realization of $\mathbf{B}$ nor a Maxwell solution. The word "Stern-Gerlach" names this conditional model, not a reported experiment; no measured value of ${\gamma }_{1\mathrm{/}2}$ enters and no empirical conclusion is drawn.

• What "force" means here. Clause (b) is a rate of change of the mean of the transverse momentum operator in a declared preparation; it is the model counterpart of the classical force $-\mathrm{\nabla }(-\mathit{\mu }\cdot \mathbf{B})$ and does not assert a classical trajectory or a definite transverse displacement of an individual particle.
• The marginal hypothesis. The support condition on ${\eta }_{+}$ is an explicit hypothesis on the preparation, not a general property of solutions; without it the general rate of clause (b) is the only identity available, since $B_z^{\mathrm{\prime }}$ enters pointwise.