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The idealized Stern-Gerlach splitting: transverse momentum separation of the two spin sectors under a window hypothesis

Physical scope

An idealized model prediction of the spin-half Zeeman model with a static windowed-gradient field: under an explicitly hypothetical containment premise (the first-coordinate marginals of both spin-sector components remain supported in the window of constant gradient during [0,tau]), the normalized sector mean transverse momenta change at rates plus and minus gamma hbar G / 2, so their difference changes by gamma hbar G tau over the time tau; the change is zero if gamma G is zero. The sector weights are conserved; the total transverse mean is conserved for equal sector weights or zero coupling-gradient product, and otherwise has rate gamma hbar G (w_plus-w_minus) / 2. The item is a hypothetical calculation, not a reported experiment: it asserts no apparatus, no beam, no counts, no detector, no measured value and no claim that the containment premise holds for realistic preparations.

Statement

Adopt the postulates P1, P2, P4, P5 and P6 and the effective magnetic-moment coupling as in The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field, for the massive spin- $\frac{1}{2}$ model with mass parameter $m > 0$ and parameter $γ : = γ_{1 / 2} \in R$ (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, Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Fix a real $B_{z} \in C^{1} (R)$ with $B_{z}$ , $B_{z}^{'}$ bounded, reals $0 < a < \infty$ and $G$ , and the field profile $B (x) = (0, 0, B_{z} (x_{1}))$ , and suppose that $B_{z}^{'} (x_{1}) = G$ for every $x_{1} \in (- a, a)$ . Let $H_{Z}$ be the Zeeman operator of the model with flow $u (t) = e^{- i t H_{Z} / ℏ} u_{0}$ for a unit vector $u_{0} \in D (H_{Z}^{2})$ , let $u_{\pm} (t) : = P_{\pm} u (t)$ be the two spin-sector components with $P_{\pm} = \frac{1}{2} (I \pm 2 S_{3} / ℏ)$ , and write $w_{\pm} : = ∥ u_{\pm} (0) ∥^{2}$ and, for each sign, let $η_{\pm} (t, x_{1}) : = \int_{R^{2}} ∣ u_{\pm}^{\pm} (t, x_{1}, x_{2}, x_{3}) ∣^{2} d x_{2} d x_{3}$ be the first-coordinate marginal density of the $\pm$ component of $u_{\pm} (t)$ , where $u_{\pm}^{\pm}$ denotes that nonzero component. Fix $τ > 0$ and adopt the following hypothetical containment premise: for each sign and every $t \in [0, τ]$ , $\int_{R ∖ (- a, a)} η_{\pm} (t, x_{1}) d x_{1} = 0.$ Then:

(i) the sector weights are conserved: $∥ u_{\pm} (t) ∥^{2} = w_{\pm}$ for every $t$ , and $⟨ S_{3} ⟩_{u (t)} = \frac{ℏ}{2} (w_{+} - w_{-})$ is independent of $t$ ;

(ii) under the containment premise, for each sign with $w_{\pm} > 0$ the normalized sector transverse mean $M_{\pm} (t) : = w_{\pm}^{- 1} ⟨ P_{x} u_{\pm} (t), u_{\pm} (t) ⟩$ is affine in $t$ on $[0, τ]$ : $\frac{d}{d t} M_{\pm} (t) = \pm \frac{γ ℏ G}{2}, M_{\pm} (τ) = M_{\pm} (0) \pm \frac{γ ℏ G τ}{2},$ so, when $w_{+} w_{-} > 0$ , their difference changes by $γ ℏ G τ$ : $M_{+} (τ) - M_{-} (τ) = M_{+} (0) - M_{-} (0) + γ ℏ G τ,$ independently of the initial means; a sector with $w_{\pm} = 0$ is trivial;

(iii) the total transverse mean evolves by the weighted rate $\frac{d}{d t} ⟨ P_{x} u (t), u (t) ⟩ = \frac{γ ℏ G}{2} (w_{+} - w_{-}) = γ G ⟨ S_{3} ⟩_{u_{0}}, t \in [0, τ],$ so for the equal-weight preparation $w_{+} = w_{-} = \frac{1}{2}$ the total transverse mean is conserved while the sector mean changes are equal and opposite; if $γ G = 0$ , both changes vanish;

(iv) Status and scope. This is an idealized hypothetical calculation: the containment premise is an explicitly assumed possible property of the evolved state, not an observed fact and not a consequence of the model; no position-space splitting, detector trace, count, calibration or measured value is asserted, no apparatus is modeled, and the historical Stern-Gerlach experiment is not described or explained by this item. The identification of the model with any real beam experiment remains a tested identification with uncertainty under Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions.

Facts & Assumptions

Given: The postulates P1, P2, P4, P5 and P6 and the magnetic-moment postulate in the stated readings; the two choice principles; the parameters $m > 0$ , $s = \frac{1}{2}$ , $γ \in R$ and the real $B_{z} \in C^{1} (R)$ with $B_{z}, B_{z}^{'}$ bounded; the reals $a > 0$ , $G$ , $τ > 0$ ; the profile $B$ ; the operator $H_{Z}$ ; the flow $u (t)$ ; the unit vector $u_{0} \in D (H_{Z}^{2})$ ; the sector projections $P_{\pm}$ ; the marginal densities $η_{\pm}$ .

[F1]

Model force theorem. For the present model and profile, with the identifications of The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field clauses (a),(b): $H_{Z}$ is self-adjoint on $W^{2, 2}$ , $S_{3}$ commutes with $H_{Z}$ on the domain, the sector weights $∥ P_{\pm} u (t) ∥^{2}$ and the mean $⟨ S_{3} ⟩_{u (t)}$ are independent of $t$ , and for every $v_{0} \in D (H_{Z}^{2})$ with flow $v (t) = e^{- i t H_{Z} / ℏ} v_{0}$ one has $\frac{d}{d t} ⟨ P_{x} v (t), v (t) ⟩ = γ ⟨ S_{3} B_{z}^{'} (Q_{x}) v (t), v (t) ⟩;$ if additionally $P_{+} v_{0} = v_{0}$ and $B_{z}^{'} = G$ on a Borel set carrying the first-coordinate marginal of $v_{0}^{+}$ , then $\frac{d}{d t} ∣_{t = 0} ⟨ P_{x} v (t), v (t) ⟩ = \frac{γ ℏ}{2} G$ ; in the model readings the operator $S_{3}$ is $\frac{ℏ}{2}$ times the Pauli $σ_{z}$ on the internal factor (The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field clauses (a),(b), Finite spin matrices clause (e), Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings clause (c)).

[F2]

Sector invariance. $P_{\pm} = \frac{1}{2} (I \pm 2 S_{3} / ℏ)$ are orthogonal projections with $P_{+} + P_{-} = I$ and $S_{3} P_{\pm} = \pm \frac{ℏ}{2} P_{\pm}$ ; since $S_{3}$ commutes with $H_{Z}$ on $D (H_{Z})$ by [F1], $P_{\pm}$ commute with $H_{Z}$ , preserve $D (H_{Z})$ and $D (H_{Z}^{2})$ , and commute with the flow, so $u_{\pm} (t) = P_{\pm} u (t) = e^{- i t H_{Z} / ℏ} P_{\pm} u_{0}$ lies in the $\pm$ spin sector for every $t$ , with $∥ u_{\pm} (t) ∥ = w_{\pm}^{1 / 2}$ (Assigned energy observable, unitary flow and the two pictures of autonomous evolution, The state space: density operators, pure and mixed states, and purity, The state space of the framework: preparations, pure rays and density operators). Moreover $P_{\pm}$ act on the internal factor while $P_{x} = P_{1}$ is the componentwise momentum in the $x_{1}$ spatial coordinate, so $P_{\pm} D (P_{x}) \subseteq D (P_{x})$ and $P_{x} P_{\pm} = P_{\pm} P_{x}$ on $D (P_{x})$ ; consequently $⟨ P_{x} u (t), u (t) ⟩ = \sum_{\pm} ⟨ P_{x} u_{\pm} (t), u_{\pm} (t) ⟩$ whenever these expectations are defined.

[F3]

Hypothetical containment premise. It is assumed that for each sign and every $t \in [0, τ]$ the density $η_{\pm} (t, \cdot)$ is supported in $(- a, a)$ ; under this premise, since $B_{z}^{'} (x_{1}) = G$ for $x_{1} \in (- a, a)$ by the window hypothesis, the identity $S_{3} B_{z}^{'} (Q_{x}) u_{\pm} (t) = \pm \frac{ℏ}{2} G u_{\pm} (t)$ holds as classes. This premise is assumed, not derived, and is labeled hypothetical (Assigned sharp observables and the outcome law of the framework, The state space of the framework: preparations, pure rays and density operators, Dimensions, SI unit conventions and quantity types of the framework).

Proof

technique · direct; decompose the state into the conserved spin sectors, apply the conditional force identity to each sector with the window premise, and integrate the resulting constant rates

Given: The data of the statement and [F1]-[F3].

1.1F1F2algebra

[Sector decomposition and conservation] By [F2] the sector components $u_{\pm} (t)$ satisfy $u_{\pm} (t) = e^{- i t H_{Z} / ℏ} u_{\pm} (0)$ with $u_{\pm} (0) = P_{\pm} u_{0} \in D (H_{Z}^{2})$ , $∥ u_{\pm} (t) ∥^{2} = w_{\pm}$ , and $⟨ S_{3} ⟩_{u (t)} = \frac{ℏ}{2} ∥ u_{+} (t) ∥^{2} - \frac{ℏ}{2} ∥ u_{-} (t) ∥^{2} = \frac{ℏ}{2} (w_{+} - w_{-})$ is independent of $t$ ; this is (i).

2.1step 1.1F1F2F3algebra

[Sector rates] Apply the general rate identity of [F1] to each sector component $u_{\pm}$ : $\frac{d}{d t} ⟨ P_{x} u_{\pm} (t), u_{\pm} (t) ⟩ = γ ⟨ S_{3} B_{z}^{'} (Q_{x}) u_{\pm} (t), u_{\pm} (t) ⟩$ ; by the numerical identity $S_{3} u_{\pm} (t) = \pm \frac{ℏ}{2} u_{\pm} (t)$ of [F2] and the premise [F3] giving $S_{3} B_{z}^{'} (Q_{x}) u_{\pm} (t) = \pm \frac{ℏ}{2} G u_{\pm} (t)$ , the right-hand side equals $\pm \frac{γ ℏ G}{2} ∥ u_{\pm} (t) ∥^{2} = \pm \frac{γ ℏ G}{2} w_{\pm}$ ; dividing by the constant sector weight $w_{\pm} > 0$ gives $\frac{d}{d t} M_{\pm} (t) = \pm \frac{γ ℏ G}{2}$ on $[0, τ]$ .

3.1step 2.1step 1.1F2algebra

[Integration and separation] Integrating the unnormalized rates of step 2.1 over $[0, τ]$ gives $⟨ P_{x} u_{\pm} (τ), u_{\pm} (τ) ⟩ = ⟨ P_{x} u_{\pm} (0), u_{\pm} (0) ⟩ \pm \frac{γ ℏ G τ}{2} w_{\pm}$ . For each sector with $w_{\pm} > 0$ , division by its conserved weight gives $M_{\pm} (τ) = M_{\pm} (0) \pm \frac{γ ℏ G τ}{2}$ ; when both weights are positive, subtracting these identities gives $M_{+} (τ) - M_{-} (τ) = M_{+} (0) - M_{-} (0) + γ ℏ G τ$ , so the difference changes by $γ ℏ G τ$ , independently of its initial value. For the total mean, [F2] gives $⟨ P_{x} u (t), u (t) ⟩ = \sum_{\pm} ⟨ P_{x} u_{\pm} (t), u_{\pm} (t) ⟩$ , so summing the unnormalized rates of step 2.1 gives $\frac{d}{d t} ⟨ P_{x} u (t), u (t) ⟩ = \frac{γ ℏ G}{2} (w_{+} - w_{-}) = γ G ⟨ S_{3} ⟩_{u_{0}}$ ; this vanishes for $w_{+} = w_{-}$ , proving (ii) and (iii).

4.1step 1.1step 2.1step 3.1F3∎

[Assembly and scope] Steps 1.1, 2.1 and 3.1 prove (i)-(iii) under the explicit premise [F3]; the premise is hypothetical, no position-space trace, count or apparatus entered, and no empirical conclusion is drawn, which is the status statement (iv).

Remarks

Normalization of the means. The mean transverse momentum of a sector is read from the outcome law of the whole preparation restricted to the sector, that is from the normalized sector mean; the statement and the proof use this reading, and the sector rate is constant under the window premise.
How strong the premise is. The containment premise constrains the prepared state and the field window jointly; for a spreading wave packet and a finite window it can fail for large $τ$ or narrow packets, and the item asserts nothing about when it holds. The physical two-beam profile of a real Stern-Gerlach magnet is not modeled here.

The idealized Stern-Gerlach splitting: transverse momentum separation of the two spin sectors under a window hypothesis

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources