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Qm Angular Momentum And Spin

1 · Prerequisites

2 · Summary

This page adds the angular momentum and spin observables to the declared massive spin- $s$ model and proves their structural properties, their spectra and outcome laws, their conservation under rotation-invariant dynamics, and the conditional coupling to a prescribed magnetic field. The vocabulary is fixed in Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings: the orbital generators $L_{i}$ are the Stone generators of the coordinate rotation groups of qm-symmetries-and-conservation read as assigned sharp observables; the spin operators $S_{i} = ℏ J_{i}$ are the fibrewise matrix actions of the internal factor assigned by P6; and the total angular momenta $J_{i}^{t o t} = L_{i} + S_{i}$ are self-adjoint sums on the orbital domain.

The supporting mathematics is local to this page and precedes every physical use. Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow proves that a self-adjoint operator perturbed by a bounded self-adjoint operator commuting with its flow is self-adjoint and that the product of the two flows is the flow of the sum with generator $i (A + B)$ . The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators constructs the $S U (2)$ lift $U_{s} (W) = U (ρ (W)) M_{s} (W)$ of the componentwise rotation action, with the explicit transported cover $ρ (W) = C^{- 1} ρ_{0} (W) C$ matching the Pauli axes to the spatial axes, proves its representation and strong-continuity properties, identifies $ρ (A_{i} (t)) = R_{i} (t)$ for the Pauli matrix subgroups $A_{i} (t)$ , proves the strong commutation of $L_{i}$ and $S_{i}$ , and identifies $U_{s} (A_{i} (t)) = e^{- i t J_{i}^{t o t} / ℏ}$ ; its kernel remark records which spin sectors carry the genuine two-valued lift. The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models proves on the finite harmonic blocks of the sphere that $(1 / ℏ) L_{i} ∣_{V_{l}}$ is unitarily equivalent to the spin- $l$ matrices, with $\sum_{i} L_{i}^{2} F = ℏ^{2} l (l + 1) F$ on the harmonic span.

The physical results are conditional on the framework postulates. Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities computes the spectrum and Born outcome law of every directional component $S_{n} = \sum_{i} n_{i} S_{i}$ , the Casimir identity $S^{2} = ℏ^{2} s (s + 1) I$ , and the explicit spin-half probabilities. Orbital angular momentum eigenstates of the massive model: spherical harmonics, the l,m quantum numbers and the Casimir value exhibits the smooth model vectors $R (∣ x ∣) h_{l m} (x) w$ with $L_{3} ψ = ℏ m ψ$ and the quadratic-form identity $\sum_{i} ∥ L_{i} ψ ∥^{2} = ℏ^{2} l (l + 1) ∥ ψ ∥^{2}$ , and reads $(l, m)$ as the orbital quantum numbers of the preparation. Conservation of total angular momentum for a spin-independent rotation-invariant bounded-potential dynamics proves that for every bounded measurable potential invariant under all rotations the lifted coordinate rotations are symmetries of the declared dynamics and that $J_{i}^{t o t}$ , $S_{i}$ and $L_{i}$ are conserved assigned sharp observables. Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator states the effective magnetic-moment coupling $μ = γ_{s} S$ and the Zeeman operator $H_{Z} = H_{0} - \sum_{i} γ_{s} B_{i} (Q) S_{i}$ with its units and the empirical status of $γ_{s}$ and $B$ , and Spin precession and the Zeeman splitting in a uniform magnetic field: Larmor angular frequency and the conservation of one component computes the uniform-field flow, the transverse precession at frequency $ω = γ_{s} B_{z}$ , the conservation of $S_{3}$ and the conjugation identities. The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field transports the conditional Stern-Gerlach force law of the magnetic page to this model's spin sectors, and Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet proves the Clebsch-Gordan decomposition of a two-spin internal factor, the ladder actions, the weight multiplicities and the uniqueness of the spin-half singlet. What the angular momentum and spin page adopts, proves and does not claim records the adopted assumptions, the proved content, the excluded topics and the empirical status of the model parameters.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow

Statement

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a nonzero complex Hilbert space (Hilbert space), let $A$ be a self-adjoint operator on $H$ with domain $D (A)$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators), and let $B$ be a bounded self-adjoint operator on $H$ (A bounded linear operator between normed spaces, Self-adjoint, positive, unitary and normal operators) with $B e^{i s A} = e^{i s A} B (s \in R),$ where $e^{i s A} = \int_{R} e^{i s λ} d E_{A} (λ)$ is the unitary group of $A$ in the spectral calculus (A self-adjoint operator generates a strongly continuous unitary group, Bounded borel pvm integral, Projection valued measure). Put $A + B$ on its natural domain $D (A + B) : = D (A) \cap D (B) = D (A)$ . Then:

(a) $A + B$ is self-adjoint;

(b) $e^{i s B} e^{i t A} = e^{i t A} e^{i s B}$ for all $s, t \in R$ ; the flows of $A$ and of the bounded $B$ commute;

(c) the family $s \mapsto P (s) : = e^{i s A} e^{i s B}$ is a strongly continuous one-parameter unitary group (Strongly continuous one-parameter unitary group), and its infinitesimal generator $G$ (Infinitesimal generator of a unitary group) is $G = i (A + B)$ : the quotient $\frac{1}{s} (P (s) x - x)$ converges as $s \to 0$ exactly for $x \in D (A)$ , with limit $i (A + B) x$ . Consequently $P (s) = e^{i s (A + B)}$ for every $s \in R$ .

Facts & Assumptions

Given: the Axiom of Choice and the Axiom of Countable Choice; a nonzero complex Hilbert space $H$ ; a self-adjoint operator $A$ ; a bounded self-adjoint operator $B$ commuting with every $e^{i s A}$ ; real $s, t, λ$ ; a unit vector or vector $x$ .

[F1]

Flow calculus. For a self-adjoint $T$ with spectral projection-valued measure $E_{T}$ and $U_{T} (s) : = e^{i s T}$ : $U_{T}$ is a strongly continuous one-parameter unitary group with $U_{T} (s)^{*} = U_{T} (- s)$ , $U_{T} (s) D (T) = D (T)$ and $T U_{T} (s) = U_{T} (s) T$ on $D (T)$ ; the quotient $\frac{1}{s} (U_{T} (s) x - x)$ converges as $s \to 0$ exactly for $x \in D (T)$ , with limit $i T x$ ; and for every bounded Borel $f : R \to C$ one has $∥ f (T) ∥ \leq \sup ∣ f ∣$ and $f (T) U_{T} (s) = U_{T} (s) f (T)$ , where $f (T) = \int f d E_{T}$ . The norm bound follows from the bounded PVM integral; the commutation follows because both factors are integrals against $E_{T}$ and the integral map is multiplicative, so $Φ_{E_{T}} (f) Φ_{E_{T}} (λ \mapsto e^{i s λ}) = Φ_{E_{T}} (f (λ) e^{i s λ}) = Φ_{E_{T}} (λ \mapsto e^{i s λ}) Φ_{E_{T}} (f)$ (Bounded borel pvm integral, Pvm integral is a star homomorphism). (A self-adjoint operator generates a strongly continuous unitary group, Bounded borel pvm integral).

[F2]

Resolvents. For $λ \neq 0$ put $r_{λ} (μ) : = (μ + i λ)^{- 1}$ and $R_{λ} : = Φ_{E_{A}} (r_{λ})$ . The bounded PVM integral gives $∥ R_{λ} ∥ \leq 1 / ∣ λ ∣$ . The spectral domain formula gives $\int μ^{2} ∣ r_{λ} (μ) ∣^{2} d E_{A, x} (μ) \leq ∥ x ∥^{2}$ , so $R_{λ} x \in D (A)$ for every $x \in H$ ; the unbounded Borel calculus product rule then gives $(A + i λ) R_{λ} = Φ_{E_{A}} (μ r_{λ} + i λ r_{λ}) = I$ on $H$ . For $y \in D (A)$ the same product rule gives $R_{λ} (A + i λ) y = Φ_{E_{A}} (r_{λ} (μ + i λ)) y = y$ . Thus $R_{λ} = (A + i λ)^{- 1}$ and $A + i λ : D (A) \to H$ is bijective. These claims use the spectral domain formula and bounded PVM norm bound (Spectral theorem for unbounded self-adjoint operators (PVM form), Bounded borel pvm integral) and the unbounded Borel calculus (Unbounded Borel functional calculus: domains, products, spectral mapping clause 2).

[F3]

Neumann series. If $R \in B (H)$ with $∥ R ∥ < 1$ , then $I - R$ is invertible and $(I - R)^{- 1} = \sum_{n \geq 0} R^{n}$ in operator norm (Neumann series and small perturbations of bounded inverses clause 1, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[F4]

Stone uniqueness. Two strongly continuous one-parameter unitary groups with the same infinitesimal generator agree at every time; equivalently $s \mapsto e^{i s T}$ is determined by the self-adjoint $T$ (Stone's theorem: unitary groups and self-adjoint generators, Infinitesimal generator of a unitary group).

[F5]

For bounded operators $S, T$ between compatible normed spaces, their composition satisfies $∥ S T ∥ \leq ∥ S ∥ ∥ T ∥$ (Composition satisfies |ST|\le|S|,|T|).

Proof

technique · direct; prove self-adjointness of $A+B$ by the Neumann series for the resolvent of $A$, differentiate the conjugation curve to obtain the commutation of the two flows, and identify the generator of the product group by comparing difference quotients, concluding with Stone's theorem

Given: The data of the statement and [F1]-[F5].

1.1givenalgebra

[Symmetry and domain of $A + B$ ] The domain $D (A + B) = D (A)$ is a dense linear subspace of $H$ because $D (A)$ is dense and $B$ is everywhere defined and linear; for $x, y \in D (A)$ , using that $A$ is symmetric and $B$ is self-adjoint and bounded, $⟨ (A + B) x, y ⟩ = ⟨ A x, y ⟩ + ⟨ B x, y ⟩ = ⟨ x, A y ⟩ + ⟨ x, B y ⟩ = ⟨ x, (A + B) y ⟩$ , so $A + B$ is densely defined and symmetric.

1.2F2F3F5algebra

[Surjectivity of $(A + B) \pm i λ$ ] Fix $λ > ∥ B ∥$ (if $B = 0$ take $λ = 1$ ); by [F2] the map $(A + i λ) : D (A) \to H$ is a bijection with inverse of norm at most $1 / λ$ , so [F5] gives $∥ (A + i λ)^{- 1} B ∥ \leq ∥ B ∥ / λ < 1$ ; [F3] applied to $- R$ makes $I + R$ invertible on $H$ , where $R : = (A + i λ)^{- 1} B$ . Since $R (H) \subseteq D (A)$ , for any $y \in D (A)$ the vector $x : = (I + R)^{- 1} y$ satisfies $x = y - R x \in D (A)$ , so $I + R$ maps $D (A)$ onto $D (A)$ . For $x \in D (A)$ , $R x \in D (A)$ and $(A + B + i λ) x = (A + i λ) (I + R) x$ ; therefore $(A + B) + i λ$ maps $D (A)$ onto $H$ , and the same argument with $- λ$ shows that $(A + B) - i λ$ maps $D (A)$ onto $H$ .

1.3F1givenalgebra

[Adjoint inclusion] Let $y \in D ((A + B)^{*})$ and put $z : = (A + B)^{*} y$ ; for every $x \in D (A)$ one has $⟨ A x, y ⟩ + ⟨ B x, y ⟩ = ⟨ x, z ⟩$ , that is $⟨ A x, y ⟩ = ⟨ x, z - B y ⟩$ because $B^{*} = B$ and $B y \in H$ , and since $A$ is self-adjoint the existence of the vector $z - B y$ representing $x \mapsto ⟨ A x, y ⟩$ forces $y \in D (A)$ with $A y = z - B y$ , so $D ((A + B)^{*}) \subseteq D (A) = D (A + B)$ with $(A + B)^{*} y = (A + B) y$ there.

1.4F1givenalgebra

[Commutation of the flows] Fix $t$ and $x \in H$ and define $g (s) : = e^{i s B} e^{i t A} e^{- i s B} x$ ; by [F1] applied to the self-adjoint operator $B$ (whose domain is all of $H$ ) the difference quotients $\frac{1}{h} (e^{i (s + h) B} y - e^{i s B} y) \to i B e^{i s B} y$ and $\frac{1}{h} (e^{- i (s + h) B} y - e^{- i s B} y) \to - i e^{- i s B} B y$ for every $y$ , so the product rule gives $g^{'} (s) = e^{i s B} (i B e^{i t A} - i e^{i t A} B) e^{- i s B} x$ , and the hypothesis $B e^{i t A} = e^{i t A} B$ makes this derivative vanish for every $s$ ; hence $g$ is constant with value $g (0) = e^{i t A} x$ , that is $e^{i s B} e^{i t A} = e^{i t A} e^{i s B}$ for all $s, t$ , which is the first identity of (b).

1.5F1algebra

[Quotient of the product group on $D (A)$ ] Put $P (s) : = e^{i s A} e^{i s B}$ and let $x \in D (A)$ ; the algebraic identity $\frac{1}{s} (P (s) x - x) = e^{i s A} \frac{1}{s} (e^{i s B} x - x) + \frac{1}{s} (e^{i s A} x - x)$ holds for every $s \neq 0$ , and as $s \to 0$ the first term converges to $i B x$ because $\frac{1}{s} (e^{i s B} x - x) \to i B x$ in norm by [F1] and $e^{i s A} \to I$ strongly, while the second converges to $i A x$ by [F1]; hence the quotient of $P$ at $x$ converges, with limit $i (A + B) x$ .

2.1step 1.1step 1.3

[Self-adjointness] Step 1.1 gives $A + B \subseteq (A + B)^{*}$ and step 1.3 gives $(A + B)^{*} \subseteq A + B$ including domains, so $A + B = (A + B)^{*}$ , which is (a).

2.2step 1.4F1algebra

[Group property and strong continuity] By step 1.4, $P (s) P (t) = e^{i s A} e^{i s B} e^{i t A} e^{i t B} = e^{i s A} e^{i t A} e^{i s B} e^{i t B} = P (s + t)$ and $P (s)^{*} = e^{- i s B} e^{- i s A} = e^{- i s A} e^{- i s B} = P (- s)$ ; each factor is strongly continuous, and the estimate $∥ P (s) x - P (s_{0}) x ∥ \leq ∥ e^{i s A} (e^{i s B} x - e^{i s_{0} B} x) ∥ + ∥ (e^{i s A} - e^{i s_{0} A}) e^{i s_{0} B} x ∥$ shows that $P$ is strongly continuous, hence a strongly continuous one-parameter unitary group.

2.3step 1.4F1algebra

[No larger domain] Let $x \in H$ and suppose that $\frac{1}{s} (P (s) x - x)$ converges, to $y \in H$ ; by step 1.4, $P (s) = e^{i s B} e^{i s A}$ , so $e^{i s A} x = e^{- i s B} P (s) x$ and the identity $\frac{1}{s} (e^{i s A} x - x) = e^{- i s B} \frac{1}{s} (P (s) x - x) + \frac{1}{s} (e^{- i s B} x - x)$ holds; the first term converges to $y$ because $e^{- i s B} \to I$ strongly and $\frac{1}{s} (P (s) x - x) \to y$ , and the second converges to $- i B x$ by [F1] applied to the self-adjoint operator $- B$ , so $\frac{1}{s} (e^{i s A} x - x)$ converges and [F1] gives $x \in D (A)$ with $y = i (A + B) x$ .

3.1step 1.5step 2.2step 2.3

[Generator identification] By step 2.2 the family $P$ is a strongly continuous one-parameter unitary group with infinitesimal generator $G$ ; steps 1.5 and 2.3 show that the quotient of $P$ at $x$ converges exactly for $x \in D (A)$ , with limit $i (A + B) x$ ; hence $D (G) = D (A)$ and $G = i (A + B)$ , which is the generator statement of (c).

4.1step 1.4step 3.1F1F4∎

[Identification of the groups] By [F1] the family $s \mapsto e^{i s (A + B)}$ is a strongly continuous one-parameter unitary group whose quotient at $x$ converges exactly for $x \in D (A + B) = D (A)$ , to $i (A + B) x$ ; by step 3.1 it has the same infinitesimal generator $i (A + B)$ as $P$ , so Stone's theorem [F4] yields $P (s) = e^{i s (A + B)}$ for every $s$ , completing (c); (b) is step 1.4.

Remarks

Why the hypothesis is a commutation with the flow. The hypothesis $B e^{i s A} = e^{i s A} B$ for all $s$ is a commutation of the bounded operator with every element of the unitary group of $A$ ; it implies, but is not assumed as, a formal commutator identity on a small domain, and it is the exact form in which the model's internal matrices commute with the free flow.
Bounded perturbation. Clause (a) is the standard self-adjointness of a self-adjoint operator perturbed by a bounded self-adjoint operator; the flow hypothesis is not needed for it, only for the product formula of (b) and the generator statement of (c).

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), the choice hypotheses of the cited spectral suppliers. Fix $ℏ > 0$ and $s \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ , put $r : = 2 s + 1$ , and let $H_{s} = L^{2} (R^{3}; C^{r})$ with inner product linear in the first argument (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions, Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument). Let $S O (3)$ be the special orthogonal group (The orthogonal group $O (3)$ and the rotation group $S O (3)$ ), let $U (R)$ for $R \in S O (3)$ be the componentwise spatial rotation action $(U (R) f)_{l} (x) : = f_{l} (R^{- 1} x) (l = 1, \dots, r)$ of The spatial rotation action on the component-valued wave-function space, and let $ρ_{0} : S U (2) \to S O (3)$ be the quaternion conjugation cover of SU(2) to SO(3) as a covering homomorphism, using the matrix identification $a + b i + c j + d k ⟼ (\begin{array}{cc} a + b i & c + d i \\ - c + d i & a - b i \end{array})$ and the imaginary basis $(i, j, k)$ of The quaternion rotation map $ρ$ : conjugation on the imaginary quaternions, the unit sphere $S^{3}$ , and quaternion lifts. For the physical coordinate axes used by the Pauli matrices fix $C : = (\begin{array}{ccc} 0 & 0 & - 1 \\ 0 & - 1 & 0 \\ - 1 & 0 & 0 \end{array}) \in S O (3), ρ (W) : = C^{- 1} ρ_{0} (W) C .$ Thus $ρ$ is the same two-sheeted cover transported to the axes $(C e_{1}, C e_{2}, C e_{3}) = (- e_{3}, - e_{2}, - e_{1})$ ; it is not the literal matrix $ρ_{0}$ in the imaginary basis. It is smooth by Covering homomorphisms of Lie groups. Let $F_{s} : S_{2 s} \to V_{s} = C^{r}$ be the unitary identification $v_{k} \mapsto e_{s - k}$ of Symmetric-power SU2 representation clause (d) (for $s = 0$ , $S_{0} = C v_{0}$ and $F_{0} v_{0} : = 1$ ), define $D_{s} : S U (2) \to U (V_{s})$ by $D_{s} (W) : = F_{s} U_{2 s} (W) F_{s}^{- 1}$ for $s > 0$ and $D_{0} (W) : = I_{C}$ , and let $S_{i}^{f i n} = ℏ J_{i}$ be the self-adjoint spin endomorphisms of $V_{s}$ of Finite spin matrices clause (a). By clauses (b),(d),(f) of the theorem the maps $D_{s}$ are strongly continuous unitary representations of $S U (2)$ . For $W \in S U (2)$ define the matrix action $M_{s} (W) \in B (H_{s})$ and the lifted operator $U_{s} (W)$ by $(M_{s} (W) f) (x) : = D_{s} (W) (f (x)), U_{s} (W) : = U (ρ (W)) M_{s} (W) (f \in H_{s}, x \in R^{3}),$ where the matrix $D_{s} (W)$ acts on the column vector $f (x) \in C^{r}$ . For any $T \in End (V_{s})$ write $M_{T} f (x) : = T f (x)$ almost everywhere, and define the spin operators on $H_{s}$ by $S_{i} : = M_{S_{i}^{f i n}}, D (S_{i}) : = H_{s} .$ These bounded self-adjoint operators, rather than the endomorphisms $S_{i}^{f i n}$ on $V_{s}$ , occur in all sums, spectral commutation claims and conjugations on $H_{s}$ below. Finally let $R_{i} (t)$ , $i = 1, 2, 3$ , be the coordinate rotation matrices and $L_{i}$ the self-adjoint coordinate-rotation generators on $H_{s}$ of Coordinate-rotation generators on the Schwartz classes and their commutators (so that $U (R_{i} (t)) = e^{- i t L_{i} / ℏ}$ ), and put $A_{i} (t) : = \cos (t / 2) I_{2} - i \sin (t / 2) σ_{i} \in S U (2)$ with the Pauli matrices $σ_{i}$ of Finite spin matrices clause (e). Then:

(a) for every $W \in S U (2)$ the operator $U_{s} (W)$ is unitary, with $U_{s} (I) = i d$ and $U_{s} (W_{1} W_{2}) = U_{s} (W_{1}) U_{s} (W_{2})$ for all $W_{1}, W_{2} \in S U (2)$ , and $W \mapsto U_{s} (W)$ is strongly continuous (Strongly continuous unitary representations, invariant linear subspaces and intertwiners);

(b) $ρ (A_{i} (t)) = R_{i} (t)$ for every $t \in R$ and every $i$ , and $t \mapsto U_{s} (A_{i} (t))$ is a strongly continuous one-parameter unitary group (Strongly continuous one-parameter unitary group);

(c) the operators $L_{i}$ and $S_{i}$ strongly commute (their spectral projection-valued measures commute; Self-adjoint, positive, unitary and normal operators, Bounded borel pvm integral), the sum $J_{i} : = L_{i} + S_{i}$ with domain $D (J_{i}) : = D (L_{i})$ is self-adjoint, and $U_{s} (A_{i} (t)) = e^{- i t J_{i} / ℏ} (t \in R);$ moreover for $W \in S U (2)$ one has the conjugation identities including domains $U_{s} (W) Q_{j} U_{s} (W)^{- 1} = \sum_{l = 1}^{3} ρ (W)_{l j} Q_{l}, U_{s} (W) P_{k} U_{s} (W)^{- 1} = \sum_{l = 1}^{3} ρ (W)_{l k} P_{l},$ where $Q_{j}, P_{k}$ are the componentwise position and momentum operators of Componentwise position and momentum operators on the component-valued wave-function space clause (a) and $ρ (W)_{l j}$ are the matrix entries of the transported cover in the standard basis; the sums mean the maximal directional realizations $Q (ρ (W) e_{j})$ and $P (ρ (W) e_{k})$ of The spatial rotation action on the component-valued wave-function space clause (b), with domains $U_{s} (W) D (Q_{j})$ and $U_{s} (W) D (P_{k})$ ;

(d) Scope. The statement is a mathematical statement about the finite-dimensional spin representation and the componentwise rotation action; it asserts no physical interpretation, no measurement, no dynamics and no value of $ℏ$ , $s$ or $W$ . The spatial action enters only through $U (R) = U (ρ (W))$ and the internal action only through $M_{s} (W)$ ; no action on a second spatial or internal factor is introduced.

Facts & Assumptions

Given: the Axiom of Countable Choice and the Axiom of Choice; the spin label $s$ , $r = 2 s + 1$ , the space $H_{s}$ ; the rotation action $U (R)$ and the covering homomorphism $ρ$ ; the identification $F_{s}$ and the representation $D_{s} = F_{s} U_{2 s} F_{s}^{- 1}$ with spin matrices $S_{i}^{f i n} = ℏ J_{i}$ and operators $S_{i} = M_{S_{i}^{f i n}}$ on $H_{s}$ ; the operators $M_{s} (W)$ and $U_{s} (W)$ ; the coordinate rotations $R_{i} (t)$ , $A_{i} (t)$ and the generators $L_{i}$ ; $W, W_{1}, W_{2} \in S U (2)$ , real $s_{0}, t$ (use $u$ for the group parameter where it is not the spin label).

[F1]

The covering and its axes. The quaternion cover $ρ_{0}$ is a smooth surjective two-sheeted covering homomorphism with kernel ${I_{2}, - I_{2}}$ . The displayed $C$ satisfies $C^{T} C = I$ , $C^{- 1} = C$ and $\det C = 1$ . Conjugation $R \mapsto C^{- 1} R C$ is a smooth automorphism of $S O (3)$ with smooth inverse, so $ρ = C^{- 1} ρ_{0} C$ has the same covering, continuity, surjectivity, multiplicativity and kernel properties. The canonical matrix identification and quaternion conjugation formula are those of SU(2) to SO(3) as a covering homomorphism, The quaternion rotation map $ρ$ : conjugation on the imaginary quaternions, the unit sphere $S^{3}$ , and quaternion lifts and The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ .

[F2]

Spatial rotation action. $U (R)$ is a unitary operator for every $R$ with $U (I) = i d$ , $U (R S) = U (R) U (S)$ and $U (R)^{- 1} = U (R^{- 1})$ ; the map $R \mapsto U (R)$ is strongly continuous on $S O (3)$ ; and $U (R) Q_{j} U (R)^{- 1} = Q (R e_{j}) = \sum_{l} R_{l j} Q_{l}$ , $U (R) P_{k} U (R)^{- 1} = P (R e_{k}) = \sum_{l} R_{l k} P_{l}$ as operator identities on the maximal directional domains defined by The spatial rotation action on the component-valued wave-function space clause (b). Explicitly, $Q (a)$ is multiplication by $a \cdot x$ on its maximal domain $D (Q (a)) = {f \in H_{s} : (a \cdot x) f \in H_{s}}$ , and $P (a) = (U^{(r)})^{- 1} M_{a \cdot p} U^{(r)}$ on its maximal domain $D (P (a)) = {f \in H_{s} : (a \cdot p) U^{(r)} f \in L^{2} (R^{3}; C^{r})}$ , where $U^{(r)}$ is the normalized Fourier unitary and $M_{a \cdot p}$ is maximal multiplication by $a \cdot p$ from that clause. In particular the conjugate domains are $U (R) D (Q_{j}) = D (Q (R e_{j}))$ and $U (R) D (P_{k}) = D (P (R e_{k}))$ , not necessarily the original individual coordinate domains (clauses (a),(b)).

[F3]

Coordinate rotation generators. For each $i$ , $U_{i} (t) : = U (R_{i} (t))$ is a strongly continuous one-parameter unitary group with $U_{i} (t) = e^{- i t L_{i} / ℏ}$ for the self-adjoint generator $L_{i}$ , and for $ψ \in S^{(r)}$ one has $L_{i} ψ = - i ℏ \sum_{j} (e_{i} \times x)_{j} D_{j} ψ = \sum_{j, k} ε_{i j k} Q_{j} P_{k} ψ$ (Coordinate-rotation generators on the Schwartz classes and their commutators clauses (a),(b), Componentwise position and momentum operators on the component-valued wave-function space).

[F4]

Internal representation. For $s > 0$ , $n = 2 s$ and $F_{s} : S_{n} \to V_{s}$ the unitary identification $v_{k} \mapsto e_{s - k}$ of Symmetric-power SU2 representation clause (d), $D_{s} (W) = F_{s} U_{n} (W) F_{s}^{- 1}$ with $U_{n} (W) = W^{\otimes n} ∣_{S_{n}}$ defines a group homomorphism $S U (2) \to U (V_{s})$ that is continuous in operator norm; for $s = 0$ , $D_{0} (W) = I_{C}$ ; and $S_{i}^{f i n} = ℏ J_{i}$ are the self-adjoint matrices of Finite spin matrices with $[S_{i}^{f i n}, S_{j}^{f i n}] = i ℏ \sum_{k} ε_{i j k} S_{k}^{f i n}$ and $\sum_{i} (S_{i}^{f i n})^{2} = ℏ^{2} s (s + 1) I$ (Symmetric-power SU2 representation clauses (b),(c),(d),(f), Finite spin matrices clauses (a),(b),(c)).

[F5]

Internal coordinate generator. For every $i$ the curve $t \mapsto A_{i} (t)$ lies in $S U (2)$ , satisfies $A_{i} (s_{0}) A_{i} (t) = A_{i} (s_{0} + t)$ and $A_{i} (0) = I_{2}$ , and the family $t \mapsto D_{s} (A_{i} (t))$ has infinitesimal generator $- i J_{i}$ : for every $v \in V_{s}$ , $\frac{1}{t} (D_{s} (A_{i} (t)) v - v) \to - i J_{i} v$ (Symmetric-power SU2 representation clauses (d),(e), Stone's theorem: unitary groups and self-adjoint generators, Infinitesimal generator of a unitary group, A self-adjoint operator generates a strongly continuous unitary group).

[F6]

Matrix actions commute with componentwise operators. For a fixed matrix $T \in End (C^{r})$ , define $(M_{T} f) (x) : = T f (x)$ . This is a bounded operator on $H_{s}$ because $∥ T f (x) ∥ \leq ∥ T ∥ ∥ f (x) ∥$ pointwise; it commutes with $U (R)$ because $T f (R^{- 1} x) = (T f) (R^{- 1} x)$ , and it commutes with $Q_{j}$ and $P_{k}$ and preserves their domains because those operators act componentwise by scalar multiplication and weak differentiation, respectively. For $W \in S U (2)$ , $M_{s} (W) = M_{D_{s} (W)}$ is unitary since $D_{s} (W)$ is unitary. These identities follow from the componentwise definitions (Componentwise position and momentum operators on the component-valued wave-function space clause (a), A bounded linear operator between normed spaces, Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces).

[F7]

Commutation with a flow. If a bounded $S \in B (H)$ commutes with every element $e^{i u T}$ of the flow of a self-adjoint $T$ , then $S$ commutes with every spectral projection $E_{T} (Λ)$ (Unitary conjugation, generator transport and strong commutation of self-adjoint observables clause (c)).

[F8]

Calculus of a strongly commuting pair. If the self-adjoint operators $A$ and $B$ have commuting spectral projection-valued measures, then $f (B) g (A) = g (A) f (B)$ for all bounded Borel $f, g$ (Bounded Borel calculus of strongly commuting self-adjoint operators clause (a), Strong spectral commutation of self-adjoint operators clause (b)).

[F9]

Product flow. If $A$ is self-adjoint and $B$ is bounded self-adjoint with $B e^{i u A} = e^{i u A} B$ for all real $u$ , then $A + B$ with domain $D (A)$ is self-adjoint and $e^{i u (A + B)} = e^{i u A} e^{i u B}$ for all $u$ (Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow clauses (a),(c)).

[F10]

Finite spectral decomposition. A normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely). Each self-adjoint $S_{i}^{f i n}$ is normal and has real eigenvalues: for a unit eigenvector $v$ , $λ = ⟨ S_{i}^{f i n} v, v ⟩$ is real. Grouping the eigenbasis by eigenvalue gives mutually orthogonal eigenprojections $P_{λ}$ with $\sum_{λ} P_{λ} = I_{V_{s}}$ and $S_{i}^{f i n} = \sum_{λ} λ P_{λ}$ ; its spectral projection on a Borel set $Σ$ is $\sum_{λ \in Σ} P_{λ}$ .

Proof

Proof technique: direct; verify unitarity and multiplicativity from the matrix action, compute the quaternion image of the coordinate subgroups, transport the commutation of the internal matrices with the spatial flow into strong spectral commutation, and identify the coordinate generators through the product-flow lemma and Stone's theorem.

Given: The data of the statement and [F1]-[F10].

1.1F1F2F4F6algebra

[Unitary and representation law] Let $W \in S U (2)$ ; the fibrewise matrix action $M_{s} (W)$ is well defined on almost-everywhere classes, linear, and isometric because $∥ D_{s} (W) f (x) ∥_{C^{r}} = ∥ f (x) ∥_{C^{r}}$ for almost every $x$ with $D_{s} (W)$ unitary, so $∥ M_{s} (W) f ∥ = ∥ f ∥$ and $M_{s} (W)$ extends to a unitary operator with inverse $M_{s} (W^{- 1})$ , and $M_{s} (W_{1}) M_{s} (W_{2}) = M_{s} (W_{1} W_{2})$ again componentwise; $U (ρ (W))$ is unitary by [F2], so $U_{s} (W)$ is unitary with $U_{s} (I) = U (I) M_{s} (I) = i d$ . Using [F6] to move $M_{s} (W_{1})$ across $U (ρ (W_{2}))$ and the multiplicativity $ρ (W_{1} W_{2}) = ρ (W_{1}) ρ (W_{2})$ of [F1] together with $D_{s} (W_{1} W_{2}) = D_{s} (W_{1}) D_{s} (W_{2})$ , one gets $U_{s} (W_{1}) U_{s} (W_{2}) = U (ρ (W_{1})) U (ρ (W_{2})) M_{s} (W_{1}) M_{s} (W_{2}) = U_{s} (W_{1} W_{2})$ .

1.2F1F3givenalgebra

[Coordinate rotations under the transported cover] Put $c = \cos (t / 2)$ and $d = \sin (t / 2)$ . In the canonical matrix identification of [F1], the matrices $A_{1} (t), A_{2} (t), A_{3} (t)$ correspond respectively to $c - d k, c - d j, c - d i$ , hence uniformly to $q = c + d φ (C e_{i})$ . For a unit axis $r$ and $v = α r + w$ with $w ⊥ r$ , quaternion multiplication $u v = - u \cdot v + u \times v$ gives $(c + d r) v (c - d r) = (c^{2} + d^{2}) α r + (c^{2} - d^{2}) w + 2 c d (r \times w) = α r + \cos t w + \sin t (r \times w) .$ Use $r = φ (C e_{i})$ and identify vectors through $φ$ . Since $C \in S O (3)$ , it preserves inner products and satisfies $(C u) \times (C v) = C (u \times v)$ (the determinant definition of the cross product); conjugating this Rodrigues formula by $C$ gives precisely the rotation about $e_{i}$ through $t$ . Therefore $ρ_{0} (A_{i} (t)) = C R_{i} (t) C^{- 1}$ and $ρ (A_{i} (t)) = R_{i} (t)$ , with exactly the coordinate convention of [F3].

1.3F4F6F7F8F10algebra

[Strong commutation of the spatial generators with internal projections] The finite spectral decomposition $S_{i}^{f i n} = \sum_{λ} λ P_{λ}$ gives $S_{i} = \sum_{λ} λ M_{P_{λ}}$ ; by [F6] the $M_{P_{λ}}$ are mutually orthogonal projections summing to $I_{H_{s}}$ . Thus $S_{i}$ is bounded self-adjoint and its spectral projections are $E_{S_{i}} (Σ) = \sum_{λ \in Σ} M_{P_{λ}} = M_{E_{S_{i}^{f i n}} (Σ)}$ . Each of these operators commutes with $U (R_{i} (u)) = e^{- i u L_{i} / ℏ}$ by [F6], hence with every $E_{L_{i}} (Λ)$ by [F7]. Both spectral measures now act on $H_{s}$ , and $E_{L_{i}} (Λ) E_{S_{i}} (Σ) = E_{S_{i}} (Σ) E_{L_{i}} (Λ)$ for all Borel $Λ, Σ$ , proving strong commutation in the sense of [F8].

1.4F1F2F4algebra

[Strong continuity of the lift] Let $W_{m} \to W$ in $S U (2)$ and $f \in H_{s}$ ; since $ρ$ is continuous by [F1], $ρ (W_{m}) \to ρ (W)$ and [F2] gives $∥ U (ρ (W_{m})) g - U (ρ (W)) g ∥ \to 0$ for every fixed $g$ ; since $∥ D_{s} (W_{m}) - D_{s} (W) ∥ \to 0$ in operator norm by [F4], $∥ M_{s} (W_{m}) f - M_{s} (W) f ∥ \leq ∥ D_{s} (W_{m}) - D_{s} (W) ∥ ∥ f ∥$ tends to $0$ ; combining, $∥ U_{s} (W_{m}) f - U_{s} (W) f ∥ \leq ∥ U (ρ (W_{m})) (M_{s} (W_{m}) f - M_{s} (W) f) ∥ + ∥ U (ρ (W_{m})) M_{s} (W) f - U (ρ (W)) M_{s} (W) f ∥ \to 0$ , so $W \mapsto U_{s} (W)$ is strongly continuous.

2.1step 1.3F4F5F6algebra

[Internal coordinate generator] By [F5] the family $t \mapsto D_{s} (A_{i} (t))$ on $V_{s}$ has infinitesimal generator $- i J_{i}$ ; the finite spectral exponential $e^{- i t S_{i}^{f i n} / ℏ}$ has the same generator $- i S_{i}^{f i n} / ℏ = - i J_{i}$ , so Stone's theorem gives $D_{s} (A_{i} (t)) = e^{- i t S_{i}^{f i n} / ℏ}$ . Lifting the finite spectral sum of step 1.3 then gives $M_{s} (A_{i} (t)) = M_{e^{- i t S_{i}^{f i n} / ℏ}} = \sum_{λ} e^{- i t λ / ℏ} M_{P_{λ}} = e^{- i t S_{i} / ℏ}$ on $H_{s}$ . This includes $s = 0$ , when both spin generators vanish.

2.2step 1.1F2F6algebra

[Conjugations of position and momentum] By [F6] the matrix action $M_{s} (W)$ commutes with $Q_{j}$ and $P_{k}$ and preserves their maximal coordinate domains, so the operator domains of the conjugates by $U_{s} (W) = U (ρ (W)) M_{s} (W)$ are $U_{s} (W) D (Q_{j}) = U (ρ (W)) D (Q_{j}) = D (Q (ρ (W) e_{j}))$ and $U_{s} (W) D (P_{k}) = U (ρ (W)) D (P_{k}) = D (P (ρ (W) e_{k}))$ , where the last equalities are [F2]. On these domains [F2] gives $U_{s} (W) Q_{j} U_{s} (W)^{- 1} = Q (ρ (W) e_{j}) = \sum_{l} ρ (W)_{l j} Q_{l}$ and $U_{s} (W) P_{k} U_{s} (W)^{- 1} = P (ρ (W) e_{k}) = \sum_{l} ρ (W)_{l k} P_{l}$ , with both sums understood as the displayed maximal directional realizations. This proves the conjugation identities of (c).

2.3step 1.3F6F8F9algebra

[Self-adjoint sum and product flow] By step 1.3 the spectral measures of $L_{i}$ and $S_{i}$ commute, so [F8] with $f (μ) = e^{- i μ t / ℏ}$ , $g (λ) = e^{- i λ u / ℏ}$ gives $e^{- i t S_{i} / ℏ} e^{- i u L_{i} / ℏ} = e^{- i u L_{i} / ℏ} e^{- i t S_{i} / ℏ}$ for all $t, u$ ; by [F6] $S_{i}$ itself also commutes with every spatial flow, so [F9] applies with $A : = L_{i}$ and the bounded self-adjoint $B : = S_{i}$ , so the sum $J_{i} = L_{i} + S_{i}$ with domain $D (L_{i})$ is self-adjoint and $e^{- i t J_{i} / ℏ} = e^{- i t L_{i} / ℏ} e^{- i t S_{i} / ℏ}$ for every $t$ .

3.1step 1.2step 2.1step 2.3F3F5algebra

[Coordinate subgroup generator] Let $t$ be real; by step 1.2, $ρ (A_{i} (t)) = R_{i} (t)$ , so by [F3] and step 2.1 $U_{s} (A_{i} (t)) = U (R_{i} (t)) M_{s} (A_{i} (t)) = e^{- i t L_{i} / ℏ} e^{- i t S_{i} / ℏ} = e^{- i t J_{i} / ℏ}$ by step 2.3; since the right-hand side is a strongly continuous one-parameter unitary group in $t$ and $A_{i}$ is multiplicative by [F5], $t \mapsto U_{s} (A_{i} (t))$ is a strongly continuous one-parameter unitary group with the displayed generator, which is (b) and the generator identity of (c).

4.1step 1.1step 1.3step 1.4step 2.2step 2.3step 3.1∎

[Assembly] Step 1.1 proves unitarity, the identity and the representation law, and step 1.4 proves strong continuity, which is (a); step 3.1 proves (b) and the generator identity of (c); step 1.3 proves the strong commutation of $L_{i}$ and $S_{i}$ , step 2.3 the self-adjointness of $J_{i}$ , and step 2.2 the conjugation identities of (c); (d) records only the scope of these mathematical assertions.

Remarks

Kernel and the two lifts. Since $D_{s} (- I_{2}) = (- 1)^{2 s} I$ by the diagonal action and Symmetric-power SU2 representation clauses (a),(b) (for $s = 0$ also $D_{0} (- I_{2}) = I$ ), one has $U_{s} (- I_{2}) = (- 1)^{2 s} i d$ . For half-integer $s$ this is $- i d \neq i d$ , so $W \mapsto U_{s} (W)$ does not factor through $ρ$ : no map $V$ with $V (ρ (W)) = U_{s} (W)$ exists, since $ρ (- I_{2}) = ρ (I_{2}) = I$ would force $V (I) = - i d$ . For integer $s$ , $- I_{2}$ acts as the identity and the action passes to the rotation group $S U (2) / {I_{2}, - I_{2}} ≅ S O (3)$ of SU(2) to SO(3) as a covering homomorphism. On a half-integer spin sector the two lifts $\pm U_{s} (W)$ of one rotation $ρ (W)$ differ by the central sign.
Units. $ℏ > 0$ is a fixed positive parameter of the statement; $L_{i}$ , $S_{i} = M_{S_{i}^{f i n}}$ and the total generator $J_{i} = L_{i} + S_{i}$ carry the unit of action. The dimensionless finite spin matrices are $S_{i}^{f i n} / ℏ$ on $V_{s}$ ; their fibrewise actions on $H_{s}$ are $S_{i} / ℏ$ . No numerical value is asserted.

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The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Work in the notation of Orbital self-adjoint generators and Sphere Laplacian diagonal domain: $S^{2}$ with its polar measure $σ$ , the Hilbert space $L^{2} (S^{2}, σ)$ with inner product linear in the first argument (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions, Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument), the complete orthonormal family $(Y_{l m})_{l \geq 0, - l \leq m \leq l}$ (Sphere ONB), the blocks $V_{l} : = span {Y_{l, - l}, \dots, Y_{l l}}$ , the finite span $H$ , the self-adjoint operators $L_{1}, L_{2}, L_{3}$ with $L_{3} = L_{z}$ and $L_{i} Y_{l m} \in V_{l}$ for all $l, m, i$ , and the diagonal sphere Laplacian $A$ with $A Y_{l m} = l (l + 1) Y_{l m}$ . Then for every $l \geq 0$ :

(a) each $L_{i}$ leaves $V_{l}$ invariant, and the restriction $L_{i} ∣_{V_{l}}$ is a self-adjoint endomorphism of the finite-dimensional inner product space $V_{l}$ (Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space), with $L_{3} Y_{l m} = ℏ m Y_{l m}$ for $- l \leq m \leq l$ ;

(b) the triple $((1 / ℏ) L_{1} ∣_{V_{l}}, (1 / ℏ) L_{2} ∣_{V_{l}}, (1 / ℏ) L_{3} ∣_{V_{l}})$ is unitarily equivalent to the spin- $l$ matrices of Finite spin matrices; in particular $\sum_{i = 1}^{3} L_{i}^{2} F = ℏ^{2} l (l + 1) F (F \in V_{l});$

The statement is a mathematical statement about the operators on $L^{2} (S^{2}, σ)$ ; no physical interpretation as angular momentum of a particle is asserted.

Facts & Assumptions

Given: The Axiom of Countable Choice; the sphere $S^{2}$ with polar measure $σ$ ; the family $(Y_{l m})$ , the blocks $V_{l}$ , the finite span $H$ ; the operators $L_{1}, L_{2}, L_{3}$ ; the diagonal operator $A$ ; a degree $l \geq 0$ ; indices $m, i$ ; a vector $F \in H$ .

[F1]

Orbital generators. Each $L_{i}$ is self-adjoint on $L^{2} (S^{2}, σ)$ ; for $F \in V_{l}$ with polynomial representative $p_{F}$ one has $L_{i} F = {r e s}_{l} (- i ℏ D_{i} p_{F})$ with $D_{i} p = (x \times \nabla p)_{i}$ , so $L_{i} V_{l} \subseteq V_{l}$ and $L_{i} H \subseteq H$ ; on $H$ one has $L_{3} = L_{z}$ with $L_{3} Y_{l m} = ℏ m Y_{l m}$ , and $L_{i} L_{j} - L_{j} L_{i} = i ℏ \sum_{k} ε_{i j k} L_{k}$ for all $i, j$ (Orbital self-adjoint generators clauses (a),(b),(c)).

[F2]

Sphere Laplacian and the basis of a block. $A$ is self-adjoint with $A Y_{l m} = l (l + 1) Y_{l m}$ , the finite span $H$ is a core of $A$ , and the $Y_{l, - l}, \dots, Y_{l l}$ are orthonormal in $L^{2} (S^{2}, σ)$ (Sphere Laplacian diagonal domain clauses (a),(b), Sphere ONB); hence $\dim V_{l} = 2 l + 1$ (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis, Orthonormal families, complete orthonormal systems and Hilbert bases).

[F3]

Finite restrictions are self-adjoint. If a self-adjoint operator $T$ on a Hilbert space leaves a finite-dimensional subspace $W$ invariant, then $T ∣_{W}$ is a self-adjoint endomorphism of $W$ : for $x, y \in W$ one has $⟨ T ∣_{W} x, y ⟩ = ⟨ T x, y ⟩ = ⟨ x, T y ⟩ = ⟨ x, T ∣_{W} y ⟩$ because $T y \in W$ and $T$ is symmetric on its domain (Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space, Symmetric, self-adjoint and essentially self-adjoint operators, Linear subspace of a vector space).

[F4]

Block decomposition of a finite angular momentum triple. If a nonzero finite-dimensional complex inner product space $V$ carries self-adjoint endomorphisms $J_{1}, J_{2}, J_{3}$ with $[J_{1}, J_{2}] = i J_{3}$ and its cyclic companions, then there are $r \geq 1$ and $j_{1}, \dots, j_{r} \in \frac{1}{2} N$ with $\sum_{ν} (2 j_{ν} + 1) = \dim V$ and an orthonormal basis $(v_{ν, k})$ of $V$ such that, with $J_{\pm} : = J_{1} \pm i J_{2}$ and $J^{2} : = J_{1}^{2} + J_{2}^{2} + J_{3}^{2}$ , $J_{3} v_{ν, k} = (j_{ν} - k) v_{ν, k}, J_{+} v_{ν, k} = \sqrt{k (2 j_{ν} - k + 1)} v_{ν, k - 1}, J_{-} v_{ν, k} = \sqrt{(k + 1) (2 j_{ν} - k)} v_{ν, k + 1}, J^{2} v_{ν, k} = j_{ν} (j_{ν} + 1) v_{ν, k};$ moreover $\dim \ker (J_{3} - M I)$ equals the number of blocks with $j_{ν} \in M + Z$ and $j_{ν} \geq M$ , and the number of blocks with $j_{ν} = M$ equals $\dim \ker (J_{3} - M I) - \dim \ker (J_{3} - (M + 1) I)$ , for $M \in \frac{1}{2} N$ ; on each block the map $v_{ν, k} \mapsto e_{j_{ν} - k}$ is a unitary equivalence of $(J_{1}, J_{2}, J_{3})$ with the spin- $j_{ν}$ matrices, and $J^{2}$ is self-adjoint and commutes with $J_{1}, J_{2}, J_{3}$ (Finite spin-block decomposition clauses (a),(b),(d), Finite spin matrices clause (b)).

[F5]

Countable sums of weights. The spectrum of $L_{3}$ on the finite space $V_{l}$ consists of the numbers $ℏ m$ , $m = - l, \dots, l$ , each with multiplicity one, because $(Y_{l m})_{m}$ is an orthonormal basis of $V_{l}$ consisting of eigenvectors by [F1], [F2] (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis, The induced length is a norm).

Proof

Proof technique: direct; restrict the self-adjoint generators to the finite blocks, invoke the finite-dimensional classification of angular momentum triples, and identify the single block and its parameter from the simple weight spectrum of $L_{3}$ .

Given: The data of the statement and [F1]-[F5].

1.1F1F2F3algebra

[Invariance and finite self-adjointness] Fix $l$ ; by [F1] $L_{i} V_{l} \subseteq V_{l}$ , and by [F3] applied to the invariant finite-dimensional subspace $V_{l} \subset L^{2} (S^{2}, σ)$ the restriction $L_{i} ∣_{V_{l}}$ is a self-adjoint endomorphism of $V_{l}$ ; by [F1] and [F2], $L_{3} Y_{l m} = ℏ m Y_{l m}$ for every $m$ with $∣ m ∣ \leq l$ , which is (a).

2.1step 1.1F1F4F5algebra

[Angular momentum relations on the block] Put $J_{i} : = (1 / ℏ) L_{i} ∣_{V_{l}}$ ; by step 1.1 the $J_{i}$ are self-adjoint endomorphisms of $V_{l}$ and by [F1] they satisfy $[J_{i}, J_{j}] = i \sum_{k} ε_{i j k} J_{k}$ on $V_{l}$ ; moreover $J^{2} : = \sum_{i} J_{i}^{2}$ commutes with each $J_{i}$ by [F4] clause (a), and the eigenvalues of $J_{3}$ on $V_{l}$ , read in units of $ℏ$ , are $- l, - l + 1, \dots, l$ , each simple by [F5].

3.1step 2.1F4algebra

[Single block of parameter $l$ ] By step 2.1 and [F4] applied to the nonzero finite-dimensional space $V_{l}$ with the triple $(J_{1}, J_{2}, J_{3})$ , there are $r \geq 1$ and parameters $j_{ν} \in \frac{1}{2} N$ whose weights $j_{ν} - k$ , $0 \leq k \leq 2 j_{ν}$ , make up the spectrum of $J_{3}$ ; since every weight of an integer parameter is an integer and every weight of $J_{3}$ is an integer by step 2.1, no parameter is a non-integer half-integer, and [F4] applied with $M = 0$ gives $r = \dim \ker J_{3} = 1$ because $0$ is a simple eigenvalue of $J_{3}$ ; applying [F4] with $M = l$ gives $1 = \dim \ker (J_{3} - l I) - \dim \ker (J_{3} - (l + 1) I)$ , so the unique parameter equals $l$ ; hence $J^{2} = l (l + 1) {i d}_{V_{l}}$ by [F4], that is $\sum_{i} L_{i}^{2} F = ℏ^{2} l (l + 1) F$ for every $F \in V_{l}$ , and the block basis of [F4] exhibits the unitary equivalence to the spin- $l$ matrices, which is (b).

4.1step 3.1F2algebra∎

[Finite span] Let $F \in H$ ; as a finite linear combination $F = \sum_{l, m} c_{l m} Y_{l m}$ , the identities of step 3.1 apply to each term because $Y_{l m} \in V_{l}$ , and $\sum_{i} L_{i}^{2} F = \sum_{l, m} c_{l m} ℏ^{2} l (l + 1) Y_{l m} = ℏ^{2} A F$ by the diagonal definition of $A$ in [F2] and the linearity of the finite sums.

Remarks

Why the block argument is not circular. The classification theorem Finite spin-block decomposition is applied to the finite restrictions $(1 / ℏ) L_{i} ∣_{V_{l}}$ ; the Casimir eigenvalue is then identified from the weight spectrum already present in Orbital self-adjoint generators, and no differential identity for $\sum_{i} D_{i}^{2}$ is assumed.
Scope. The identity concerns the operators on $L^{2} (S^{2}, σ)$ and their finite harmonic span; no domain statement for a maximal operator $\sum_{i} L_{i}^{2}$ on $L^{2} (R^{3})$ is made here.

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Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings

Physical scope

The angular momentum readings of the massive spin-s particle model: the orbital generators of the spatial coordinate rotations, the internal spin matrices of the spin assignment P6, and their self-adjoint sum, each read as an assigned sharp observable of the framework with units of action, together with the declaration that the internal factor is not a spatial object. The item fixes terminology and readings of P1, P2, P4, P5 and P6; it asserts no dynamics, no magnetic coupling, no apparatus or detection model, no spin-orbit or relativistic structure and no empirical identification of any laboratory quantity with the operators named.

Definition

(a) Adopted readings. 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 identical-particle and spin postulate P6, all read for the spin- $s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m > 0$ , spin label $s \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ and $r = 2 s + 1$ (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, The state space of the framework: preparations, pure rays and density operators, Assigned sharp observables and the outcome law of the framework). Let $H_{s} = L^{2} (R^{3}; C^{r})$ be the model's state space with componentwise operators $Q_{j}, P_{k}$ , and let $S^{(r)}$ be its componentwise Schwartz class (Componentwise position and momentum operators on the component-valued wave-function space). Units are those of Dimensions, SI unit conventions and quantity types of the framework; the unit of angular momentum is the unit of action (Js).

(b) Orbital angular momentum. For $i = 1, 2, 3$ let $L_{i}$ be the self-adjoint operator on $H_{s}$ of Coordinate-rotation generators on the Schwartz classes and their commutators, the Stone generator of the coordinate rotation group $t \mapsto U (R_{i} (t))$ , so that $U (R_{i} (t)) = e^{- i t L_{i} / ℏ}$ and, on the Schwartz class, $L_{i} ψ = \sum_{j, k = 1}^{3} ε_{i j k} Q_{j} P_{k} ψ$ (Coordinate-rotation generators on the Schwartz classes and their commutators clause (b), Strongly continuous one-parameter unitary group). The model reads $(L_{i}, J s)$ as an assigned sharp observable, called the $i$ -th orbital angular momentum of the model, with outcome law $μ_{ρ}^{L_{i}}$ in a preparation $ρ$ (Assigned sharp observables and the outcome law of the framework clauses (a),(b), The state space: density operators, pure and mixed states, and purity). The operators $L_{i}$ act on the spatial factor only; their reading on the angular degrees of freedom is the one supplied by the sphere operators Orbital self-adjoint generators and Sphere ONB, and the finite-blocks identity $\sum_{i} L_{i}^{2} F = ℏ^{2} l (l + 1) F$ on the harmonic span is the content of a separate lemma of this page (The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models).

(c) Spin angular momentum. Let $S_{i} = ℏ J_{i}$ be the internal operators of The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (c), the matrix actions $(S_{i} f)_{l} = ℏ \sum_{m} (J_{i})_{l m} f_{m}$ of the self-adjoint matrices of Finite spin matrices. The model reads $(S_{i}, J s)$ as an assigned sharp observable, the $i$ -th spin angular momentum; on the internal factor, $[S_{i}, S_{j}] = i ℏ \sum_{k = 1}^{3} ε_{i j k} S_{k}, S^{2} : = S_{x}^{2} + S_{y}^{2} + S_{z}^{2} = ℏ^{2} s (s + 1) I$ (Finite spin matrices clauses (a),(b),(c), Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii)). The ladder operators $S_{\pm} : = S_{x} \pm i S_{y}$ and the finite projections on the basis vectors $e_{m}$ are structural objects of the internal factor, not assigned sharp observables. They satisfy $S_{+}^{*} = S_{-}$ ; for $s > 0$ they are not self-adjoint, since $S_{+} e_{- s} = ℏ \sqrt{2 s} e_{- s + 1} \neq 0$ whereas $S_{-} e_{- s} = 0$ , while for $s = 0$ both vanish and are self-adjoint. For $s = \frac{1}{2}$ one has $S_{i} = \frac{ℏ}{2} σ_{i}$ in the Pauli matrices (Finite spin matrices clause (e)). The internal factor is a finite-dimensional bookkeeping factor: it is not a spatial coordinate, not a rotating body and not a relativistic spinor, and the model assigns no position or worldline to it.

(d) Total angular momentum. For each $i$ let $J_{i}^{t o t} : = L_{i} + S_{i}, D (J_{i}^{t o t}) : = D (L_{i}),$ the self-adjoint sum of The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators clause (c) (well defined because $S_{i}$ is bounded and strongly commutes with $L_{i}$ ; Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow clause (a), Symmetric, self-adjoint and essentially self-adjoint operators). The model reads $(J_{i}^{t o t}, J s)$ as an assigned sharp observable, the $i$ -th total angular momentum; its flow is the coordinate subgroup $t \mapsto U_{s} (A_{i} (t))$ of the lifted rotation action, using the physical-axes cover $ρ (W) = C^{- 1} ρ_{0} (W) C$ defined in that lemma (rather than the literal quaternion-coordinate cover $ρ_{0}$ ), $U_{s} (A_{i} (t)) = e^{- i t J_{i}^{t o t} / ℏ}$ (The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators clauses (b),(c)).

(e) Status and scope. Clauses (a)-(d) fix the angular momentum vocabulary of this page within P1, P2, P4, P5 and P6 and the cited readings. No new physical assumption is added; in particular no magnetic coupling, no spin-orbit interaction, no SU(2) gauge structure and no anomaly is introduced. Whether a laboratory angular momentum measurement is described by one of the assigned pairs $(L_{i}, J s)$ , $(S_{i}, J s)$ or $(J_{i}^{t o t}, J s)$ is a tested identification with uncertainty under the preparation and apparatus postulate P7, not a consequence of this definition; no measurement procedure, apparatus, resolution or empirical value is supplied here. Antiunitary symmetries, superselection rules and symmetries of preparations as opposed to dynamics are outside the definition.

Remarks

Why the sum is a definition and not a theorem. The operator identity $J_{i}^{t o t} = L_{i} + S_{i}$ is self-adjoint on $D (L_{i})$ because $S_{i}$ is bounded self-adjoint; the adjoint-domain argument in Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow Proof steps 1.1, 1.3 and 2.1 uses no commutation hypothesis for this conclusion. Strong commutation identifies the flow of the sum with the product of the two flows; that commutation and product-flow identity are proved in The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators, and this definition only names the resulting assigned observable.
Orbital and spin are distinct. $L_{i}$ acts on the spatial factor and $S_{i}$ on the internal factor; no term of this definition identifies the spin with a rotation of a spatial body, and the page proves no spin-orbit coupling.

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Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator

Physical scope

The effective magnetic-moment coupling of the framework's massive spin-s particle model in a declared inertial frame, for a prescribed real bounded measurable magnetic field profile and a declared real gyromagnetic parameter gamma_s: the magnetic moment mu = gamma_s S with unit J T^-1, the Zeeman energy operator H_Z on the domain of the free operator, its units and the autonomous unitary evolution assigned to it by P4, and the status of gamma_s and of the static field profile as empirical or externally prescribed model inputs. The item asserts no Maxwell equation, no source model, no derivation of gamma_s from charge and mass, no spin-orbit or relativistic correction, no time-dependent or open-system dynamics and no measured value; it is a model assumption adopted on this page only.

Postulate

Adopt as a physical assumption of the framework, for the declared massive spin- $s$ particle model of The massive spin-s particle model of the framework: state space, operators, units and frame readings and only within the scope stated below, the following effective magnetic-moment coupling (Dimensions, SI unit conventions and quantity types of the framework, 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).

(i) Gyromagnetic parameter and magnetic moment. For the declared spin label $s$ assign a real gyromagnetic parameter $γ_{s}$ , with unit $s^{- 1} T^{- 1}$ , and define the magnetic moment of the model as the internal-factor operator $μ : = γ_{s} S = (γ_{s} S_{1}, γ_{s} S_{2}, γ_{s} S_{3}),$ with unit $J T^{- 1}$ . Here $S = (S_{1}, S_{2}, S_{3})$ are the spin operators of Finite spin matrices read on the internal factor as in The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (c). The parameter $γ_{s}$ is an empirical model parameter of the species: it is part of the declared model data, with uncertainty, and it is not derived from $s$ , from the mass parameter $m$ or from the charge of the particle; no relation between $γ_{s}$ and any other species parameter is asserted by this postulate.

(ii) Prescribed field and Zeeman energy operator. Let $B : R^{3} \to R^{3}$ be a real bounded measurable vector field, the prescribed magnetic field profile, with components $B_{i}$ in tesla, entering the model as externally prescribed data (The state space of the framework: preparations, pure rays and density operators, Componentwise position and momentum operators on the component-valued wave-function space); in the declared inertial frame the model adopts the Zeeman energy operator $H_{Z} : = H_{0} + M_{W}, H_{0} : = \frac{1}{2 m} \sum_{k = 1}^{3} P_{k}^{2}, M_{W} f : = - \sum_{i = 1}^{3} γ_{s} B_{i} (\cdot) (S_{i} f) (\cdot),$ on the componentwise domain $D (H_{Z}) : = W^{2, 2} (R^{3}; C^{r})$ , the domain of $H_{0}$ (componentwise, Componentwise position and momentum operators on the component-valued wave-function space; $\sum_{k} P_{k}^{2}$ is the componentwise free operator with that domain by Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b) applied to each component). The added term $M_{W}$ is bounded with $∥ M_{W} ∥ \leq ∣ γ_{s} ∣ \sum_{i} ∥ B_{i} ∥_{\infty} \max_{i} ∥ S_{i} ∥$ ; its matrix value is Hermitian almost everywhere since $B_{i}$ and $γ_{s}$ are real and each finite spin matrix is self-adjoint. Integrating the fibrewise inner-product identity gives $M_{W}^{*} = M_{W}$ on $H_{s}$ . For completeness, if $y \in D ((H_{0} + M_{W})^{*})$ and $z = (H_{0} + M_{W})^{*} y$ , then $⟨ H_{0} x, y ⟩ = ⟨ x, z - M_{W} y ⟩$ for every $x \in D (H_{0})$ . Self-adjointness of $H_{0}$ gives $y \in D (H_{0})$ and $H_{0} y = z - M_{W} y$ ; the reverse adjoint inclusion follows from symmetry. Thus $H_{Z} = H_{Z}^{*}$ on the unchanged domain (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators), with unit joule, and the assignment $t \mapsto e^{- i t H_{Z} / ℏ}$ is a strongly continuous one-parameter unitary group read in the autonomous-dynamics postulate P4 (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, State and composition postulate P1: Hilbert spaces, preparations and composite systems, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Assigned sharp observables and the outcome law of the framework). The profile $B$ is a model input; the items of this page assert neither the Maxwell equations nor a source or boundary realization of $B$ , and the boundedness of $B$ is an explicit hypothesis, not a general property of magnetic fields.

(iii) What is assumed and what is not. (a) The coupling $- μ \cdot B (Q)$ is an effective interaction: it is adopted, not derived from the spin representation P6, from the minimal-coupling postulate P5a of the framework or from classical electromagnetism. (b) No value of $γ_{s}$ and no comparison with measured gyromagnetic ratios or $g$ -factors is made here. (c) No spin-orbit coupling, no fine or hyperfine structure, no relativistic correction and no radiation reaction is included. (d) A time-dependent field profile requires its own domain and evolution hypotheses and is outside this postulate. (e) The internal factor is not a spatial current distribution and $μ$ is not asserted to arise from a rotating charge distribution.

(iv) Status. This postulate is an adopted model assumption with the physical scope above; it is motivated by and formulated after the cited sources, which treat the coupling as a model input, and no deduction of it from the framework's other postulates is claimed.

Remarks

Why a bounded profile. The domain statement is that of the free operator on $W^{2, 2}$ ; it does not cover linearly growing fields, point dipoles or fields whose components leave $L^{\infty}$ , each of which would require a separate form-domain or resolvent analysis.
Units check. $γ_{s}$ in $s^{- 1} T^{- 1}$ times $S_{i}$ in $J s$ gives $J T^{- 1}$ for $μ_{i}$ , and $μ_{i} B_{i}$ in $J T^{- 1} \cdot T = J$ ; the phase $H_{Z} t / ℏ$ is dimensionless. The parameter $γ_{s}$ is read in the conventions of Dimensions, SI unit conventions and quantity types of the framework; no numerical value is asserted and, in particular, no exact value of any $g$ -factor is assumed.

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Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities

Physical scope

A conditional consequence of the state, observable and kinematics postulates P1, P2, P5 and of the internal spin assignment P6 for the massive spin-s model: the spectra and Born outcome laws of the explicitly assigned directional spin observables n·S, the assigned squared-spin observable with operator S^2 = hbar^2 s(s+1) I and, for s = 1/2, the explicit spin-up/spin-down probabilities. The item concerns the internal factor only; it asserts no dynamics, no magnetic coupling, no apparatus model, no measured spin value and no empirical identification of the operators.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-particle model postulate P5 and the internal-factor assignment of the spin postulate P6, read for the massive spin- $s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m > 0$ , spin label $s \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ and $r = 2 s + 1$ , together with 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, 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, Assigned sharp observables and the outcome law of the framework), and assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H_{s} = L^{2} (R^{3}; C^{r})$ , let $μ_{ψ}^{A}$ be the outcome law of an assigned sharp observable $A$ in the pure preparation of a unit vector $ψ$ , and for a unit vector $n = (n_{1}, n_{2}, n_{3}) \in R^{3}$ put $S_{n} : = \sum_{i} n_{i} S_{i}$ , the matrix action of the self-adjoint matrix $ℏ n \cdot J$ on the internal factor. For every unit $n$ , read $(S_{n}, J s)$ as an assigned sharp observable under P2. Put $S^{2} : = S_{x}^{2} + S_{y}^{2} + S_{z}^{2}$ and, for its outcome-law claim in (b), read $(S^{2}, (J s)^{2})$ as an assigned sharp observable under P2. Then:

(a) for every unit $n$ the operator $S_{n}$ is a bounded self-adjoint operator on $H_{s}$ with spectrum $σ (S_{n}) = ℏ {- s, - s + 1, \dots, s}$ , and for each $m \in {- s, \dots, s}$ the eigenspace $\ker (S_{n} - ℏ m)$ is the range of the fibrewise orthogonal projection $P_{n, m}$ onto the eigenline of the matrix $n \cdot J$ with eigenvalue $m$ ; consequently, for every unit $ψ \in H_{s}$ , $μ_{ψ}^{S_{n}} ({ℏ m}) = ∥ P_{n, m} ψ ∥^{2} = ⟨ P_{n, m} ψ, ψ ⟩ (m = - s, \dots, s),$ and the law is concentrated on the finite set $ℏ {- s, \dots, s}$ ;

(b) the Casimir operator $S^{2}$ equals $ℏ^{2} s (s + 1) I$ as a bounded operator, so for every preparation $ρ$ the law of the assigned sharp observable $(S^{2}, (J s)^{2})$ is the Dirac law at $ℏ^{2} s (s + 1)$ , and one has the commutation relations $[S_{i}, S_{j}] = i ℏ \sum_{k} ε_{i j k} S_{k}$ and $[S^{2}, S_{i}] = 0$ on $H_{s}$ , componentwise on the internal factor;

(c) for $s = \frac{1}{2}$ , every unit $n$ and every unit $ψ \in H_{1 / 2}$ one has $P_{n, \pm} = \frac{1}{2} (I \pm n \cdot σ), μ_{ψ}^{S_{n}} ({\pm \frac{ℏ}{2}}) = \frac{1}{2} (1 \pm ⟨ (n \cdot σ) ψ, ψ ⟩),$ where $n \cdot σ$ is applied fibrewise; in particular, $μ_{ψ}^{S_{n}} ({ℏ / 2}) + μ_{ψ}^{S_{n}} ({- ℏ / 2}) = 1$ and each number lies in $[0, 1]$ ;

(d) Scope. Clauses (a)-(c) are conditional consequences of P1, P2, P5 and P6, the sharp-observable assignments stated above and the cited finite-dimensional mathematics; they concern the internal factor of the model. They assert no dynamics, no magnetic coupling, no measurement or apparatus model, no joint measurement of two spin components, no value of $ℏ$ and no empirical spin value. In particular the statements do not identify the operators $S_{n}$ with any laboratory spin measurement, and they contain no comparison with data.

Facts & Assumptions

Given: The postulates P1, P2, P5 and P6 in the stated readings; the Axiom of Choice and the Axiom of Countable Choice; the spin-s model with parameters $m > 0$ , $s$ and $r = 2 s + 1$ ; the internal matrices $J_{i}$ and the spin operators $S_{i} = ℏ J_{i}$ ; a unit vector $n \in R^{3}$ ; the assigned pairs $(S_{n}, J s)$ and $(S^{2}, (J s)^{2})$ of the Statement; the operators $S_{n}$ and $S^{2}$ ; a unit vector $ψ \in H_{s}$ ; a preparation $ρ$ ; indices $i, j, k$ and eigenvalues $λ$ of a self-adjoint operator.

[F1]

Finite spin matrices. On $V_{s} = C^{r}$ the matrices $J_{i}$ are self-adjoint with $[J_{i}, J_{j}] = i \sum_{k} ε_{i j k} J_{k}$ , $J^{2} = J_{x}^{2} + J_{y}^{2} + J_{z}^{2} = s (s + 1) I$ , $J_{\pm} = J_{x} \pm i J_{y}$ , $[J^{2}, J_{i}] = 0$ , and the basis $(e_{m})$ satisfies $J_{z} e_{m} = m e_{m}$ , $J_{\pm} e_{m} = \sqrt{(s \mp m) (s \pm m + 1)} e_{m \pm 1}$ ; for $s = \frac{1}{2}$ one has $J_{i} = \frac{1}{2} σ_{i}$ with $σ_{i} σ_{j} + σ_{j} σ_{i} = 2 δ_{i j} I$ (Finite spin matrices clauses (a),(b),(c),(d),(e)).

[F2]

Finite angular momentum classification. If a nonzero finite-dimensional complex inner product space $V$ carries self-adjoint endomorphisms $A_{1}, A_{2}, A_{3}$ with $[A_{1}, A_{2}] = i A_{3}$ and its cyclic companions, then $V$ is an orthogonal direct sum of blocks on each of which the triple is unitarily equivalent to a spin- $j$ model with $A^{2} = j (j + 1) I$ (Finite spin-block decomposition clauses (b),(c), Finite spin matrices).

[F3]

Model operators as matrix actions. The model's $S_{n}$ is the fibrewise matrix action $(S_{n} f) (x) = ℏ (n \cdot J) f (x)$ on $H_{s} = L^{2} (R^{3}; C^{r})$ of a self-adjoint matrix, hence bounded self-adjoint with $∥ S_{n} ∥ \leq ℏ ∥ n \cdot J ∥$ (The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (c), Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings clause (c), A bounded linear operator between normed spaces, Self-adjoint, positive, unitary and normal operators).

[F4]

Eigenvalues and their spectral weights. If $λ$ is an eigenvalue of a self-adjoint operator $A$ with spectral projection-valued measure $E_{A}$ , then $E_{A} ({λ})$ is the orthogonal projection onto $\ker (A - λ)$ ; for the pure preparation of a unit vector $ψ$ one has $μ_{ψ}^{A} ({λ}) = ∥ E_{A} ({λ}) ψ ∥^{2}$ (Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a),(b), Assigned sharp observables and the outcome law of the framework clause (b)).

[F5]

Observable assignments and outcome laws. The model's coordinate angular momentum operators $L_{i}$ and $S_{i}$ are the assigned sharp observables of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings clauses (b),(c) with unit $J s$ . The directional and squared-spin observable assignments used here are stated in the theorem's Statement. The outcome law of each assigned sharp observable is that of P2 in the reading of Assigned sharp observables and the outcome law of the framework clause (b), and the pure preparations are those of The state space: density operators, pure and mixed states, and purity (The state space of the framework: preparations, pure rays and density operators).

Proof

technique · direct; for an arbitrary direction realize the rotated triple and apply the finite angular momentum classification to obtain a single spin-$s$ block, then read the eigenspaces as ranges of fibrewise projections and compute the law with the atom formula; the spin-half case is evaluated in the Pauli matrices

Given: The data of the statement and [F1]-[F5].

1.1F1F2algebra

[Directional component is a single spin- $s$ block] Fix a unit $n$ and choose an oriented orthonormal frame $(e^{(1)}, e^{(2)}, e^{(3)})$ of $R^{3}$ with $e^{(1)} = n$ ; put $A_{i} : = \sum_{l} e_{l}^{(i)} J_{l}$ for $i = 1, 2, 3$ ; expanding the commutator with [F1] gives $[A_{i}, A_{j}] = \sum_{l, m} e_{l}^{(i)} e_{m}^{(j)} [J_{l}, J_{m}] = i \sum_{k} (\sum_{l, m} ε_{l m k} e_{l}^{(i)} e_{m}^{(j)}) J_{k} = i \sum_{k} (e^{(i)} \times e^{(j)})_{k} J_{k}$ , which equals $i \sum_{k} ε_{i j k} A_{k}$ , including zero for $i = j$ and the negative sign for reversed indices, and $\sum_{i} A_{i}^{2} = \sum_{k} J_{k}^{2} = s (s + 1) I$ by the orthonormality of the frame; by [F2] applied to the triple $(A_{1}, A_{2}, A_{3})$ there is a decomposition into blocks with parameters $j_{ν}$ satisfying $j_{ν} (j_{ν} + 1) = s (s + 1)$ on each block, so $j_{ν} = s$ and every block has dimension $2 s + 1 = \dim V_{s}$ ; hence there is exactly one block and $A_{1} = n \cdot J$ has the simple eigenvalues $s, s - 1, \dots, - s$ , which is the spectrum statement of (a) at the matrix level.

2.1step 1.1F3F4F5algebra

[Fibrewise spectral measure and law of $S_{n}$ ] By step 1.1 the matrix $n \cdot J$ has the simple eigenvalues $m = - s, \dots, s$ ; let $P_{n, m}$ be the fibrewise action on $H_{s}$ of the orthogonal projection of $C^{r}$ onto the eigenline of $n \cdot J$ with eigenvalue $m$ ; each $P_{n, m}$ is an orthogonal projection on $H_{s}$ , $P_{n, m} P_{n, m^{'}} = 0$ for $m \neq m^{'}$ , $\sum_{m} P_{n, m} = I$ fibrewise, and $S_{n} = \sum_{m} ℏ m P_{n, m}$ as bounded operators; consequently $\ker (S_{n} - ℏ m) = ran P_{n, m}$ because a class $f$ satisfies $ℏ (n \cdot J) f (x) = ℏ m f (x)$ for almost every $x$ exactly when $f (x)$ lies in the $m$ -eigenline almost everywhere; by [F4] applied to the self-adjoint $S_{n}$ of [F3], $E_{S_{n}} ({ℏ m}) = P_{n, m}$ and $μ_{ψ}^{S_{n}} ({ℏ m}) = ∥ P_{n, m} ψ ∥^{2} = ⟨ P_{n, m} ψ, ψ ⟩$ for every unit $ψ$ ; since $\sum_{m} P_{n, m} = I$ , the law is concentrated on $ℏ {- s, \dots, s}$ , which is (a).

3.1step 2.1F1F3F5algebra

[Coordinate components, Casimir and commutators] Applying step 2.1 with $n$ a standard basis vector gives clause (a) for the coordinate components $S_{i}$ ; by [F1] the matrices satisfy $[J_{i}, J_{j}] = i \sum_{k} ε_{i j k} J_{k}$ and $J^{2} = s (s + 1) I$ , which pass to the fibrewise actions: $[S_{i}, S_{j}] = i ℏ \sum_{k} ε_{i j k} S_{k}$ and $S^{2} = ℏ^{2} J^{2} = ℏ^{2} s (s + 1) I$ as bounded operators, so $[S^{2}, S_{i}] = 0$ ; under the assigned pair $(S^{2}, (J s)^{2})$ stated in the Statement, P2 gives the outcome law of $S^{2}$ from its spectral measure, which is the Dirac law at $ℏ^{2} s (s + 1)$ because $S^{2}$ is scalar. This is (b).

3.2step 2.1F1F3F4algebra

[Spin-half probabilities] Let $s = \frac{1}{2}$ and let $n$ be unit; by [F1], $(n \cdot σ)^{2} = \sum_{i, j} n_{i} n_{j} σ_{i} σ_{j} = \sum_{i, j} n_{i} n_{j} δ_{i j} I = I$ , so $n \cdot σ$ is a self-adjoint involution with eigenvalues $\pm 1$ , and $P_{\pm} : = \frac{1}{2} (I \pm n \cdot σ)$ are orthogonal projections with $P_{+} + P_{-} = I$ and $(n \cdot σ) P_{\pm} = \pm P_{\pm}$ ; hence $P_{\pm}$ are the eigenline projections of $n \cdot J = \frac{1}{2} n \cdot σ$ for the eigenvalues $\pm \frac{1}{2}$ , and step 2.1 gives $μ_{ψ}^{S_{n}} ({\pm \frac{ℏ}{2}}) = ∥ P_{\pm} ψ ∥^{2} = ⟨ P_{\pm} ψ, ψ ⟩ = \frac{1}{2} (1 \pm ⟨ (n \cdot σ) ψ, ψ ⟩)$ , which lies in $[0, 1]$ because $P_{\pm}$ are projections and sums to $1$ ; this is (c).

4.1step 2.1step 3.1step 3.2F5∎

[Assembly and scope] Step 2.1 proves (a), step 3.1 proves (b) and step 3.2 proves (c); clauses (a)-(c) used only the internal matrices, the readings of [F5] and the atom formula [F4], no dynamics, coupling, apparatus or empirical input entered, which is the scope statement (d).

Remarks

Simplicity of the directional eigenvalues. The simplicity of the eigenvalues of $n \cdot J$ is a statement about the internal matrices; the corresponding eigenspaces of $S_{n}$ on $H_{s}$ are infinite-dimensional (spatial functions times one internal vector), and clause (a) records the Born weight of the fibrewise projection, not a finite-dimensional multiplicity.
No joint law. Nothing in this theorem asserts a joint outcome law of two different components $S_{n}$ and $S_{n^{'}}$ with $n$ not parallel to $n^{'}$ ; the incompatibility of the components is a separate result of the uncertainty page, and the present item uses no joint measurement model.

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Orbital angular momentum eigenstates of the massive model: spherical harmonics, the l,m quantum numbers and the Casimir value

Physical scope

A conditional consequence of P1, P2, P5 and P6 for the massive spin-s model and its orbital generators: for a smooth compactly supported radial profile times a spherical harmonic, the model vector is an eigenvector of the third orbital generator with eigenvalue hbar m, the Casimir quadratic form takes the value hbar^2 l(l+1), and the orbital quantum numbers are those of the spherical harmonic family. The statement concerns the spatial factor only, uses no dynamics and no measurement model, and asserts no spectral decomposition of the full spatial Hilbert space, no hydrogen spectrum and no empirical result.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-particle model postulate P5 and the internal-factor assignment of the spin postulate P6, read for the massive spin- $s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings 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, 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 ( ${A C}_{ω}$ )). Fix $l \geq 0$ , $m \in {- l, \dots, l}$ , a unit vector $w \in C^{r}$ and a complex-valued $R \in C_{c}^{\infty} ((0, \infty))$ , and let $h_{l m}$ be the harmonic homogeneous polynomial of degree $l$ with $h_{l m} ∣_{S^{2}} = Y_{l m}$ of Harmonic polynomial dimension clause (c) (Associated Legendre function normalization). Define the model vector $ψ (x) : = R (∣ x ∣) h_{l m} (x) w (x \in R^{3}, x \neq 0), ψ (0) : = 0.$ Then:

(a) $ψ$ is a well-defined class in the model's state space $H_{s}$ and in fact $ψ \in C_{c}^{\infty} (R^{3}; C^{r}) \subseteq S^{(r)}$ ; it lies in $D (L_{i})$ for each $i$ , and for $x \neq 0$ $(L_{i} ψ) (x) = R (∣ x ∣) (- i ℏ D_{i} h_{l m}) (x) w, D_{i} h_{l m} = (x \times \nabla h_{l m})_{i},$ so that $- i ℏ D_{i} h_{l m}$ is the polynomial representative of the sphere function $L_{i}^{s p h} Y_{l m}$ of Orbital self-adjoint generators; in particular $L_{3} ψ = ℏ m ψ$ ;

(b) with $∥ R ∥_{l}^{2} : = \int_{0}^{\infty} ∣ R (r) ∣^{2} r^{2 l + 2} d r$ one has $∥ ψ ∥^{2} = ∥ R ∥_{l}^{2}$ and $∥ L_{i} ψ ∥^{2} = ∥ R ∥_{l}^{2} ∥ L_{i}^{s p h} Y_{l m} ∥^{2}, \sum_{i = 1}^{3} ∥ L_{i} ψ ∥^{2} = ℏ^{2} l (l + 1) ∥ ψ ∥^{2},$ so the Casimir quadratic form of the model's orbital generators on $ψ$ takes the value $ℏ^{2} l (l + 1)$ ;

(c) consequently, for the pure preparation of the unit vector $ψ / ∥ ψ ∥$ (for $R \equiv̸ 0$ ), the mean and the second moment of the outcome law of $(L_{3}, J s)$ are finite and equal $ℏ m$ and $ℏ^{2} m^{2}$ , so the variance vanishes; for each $i$ the mean and second moment of the law of $L_{i}$ are finite and the second moment equals $∥ L_{i}^{s p h} Y_{l m} ∥^{2}$ ; the model reads $(l, m)$ as the orbital angular momentum quantum numbers of the preparation, in the sense of the eigenvalue of the angular momentum-squared form and the eigenvalue of $L_{3}$ ;

(d) Scope. Clauses (a)-(c) are conditional consequences of P1, P2, P5, P6 and the cited mathematics, and concern the spatial factor of the massive model. The item asserts no dynamics, no measurement or apparatus model, no joint sharp values of two different components $L_{i}, L_{j}$ , no empirical value and no spectral resolution of $L^{2} (R^{3})$ : the identity $\sum_{i} L_{i}^{2} = ℏ^{2} l (l + 1)$ is a quadratic-form statement on the displayed smooth vectors and is not read as a statement about a self-adjoint operator $\sum_{i} L_{i}^{2}$ on the full spatial Hilbert space.

Facts & Assumptions

Given: The postulates P1, P2, P5 and P6 in the stated readings; the Axiom of Choice and the Axiom of Countable Choice; the model with parameters $m > 0$ , $s$ , $r = 2 s + 1$ ; the harmonic polynomial $h_{l m}$ and the sphere functions $Y_{l m}$ ; the radial function $R$ ; the unit vector $w$ ; the vector $ψ$ ; indices $i, k$ .

[F1]

Harmonic polynomial and spherical harmonic. For $l \geq 0$ and $- l \leq m \leq l$ the polynomial $h_{l m}$ is homogeneous of degree $l$ , harmonic, nonzero, with $h_{l m} ∣_{S^{2}} = Y_{l m}$ , and $D_{3} h_{l m} = i m h_{l m}$ where $D_{3} = x_{1} \partial_{2} - x_{2} \partial_{1}$ ; moreover $L_{3}^{s p h} Y_{l m} = ℏ m Y_{l m}$ and $L_{i}^{s p h} Y_{l m} = {r e s}_{l} (- i ℏ D_{i} h_{l m})$ for $D_{i} p = (x \times \nabla p)_{i}$ (Harmonic polynomial dimension clause (c), Orbital self-adjoint generators clauses (a),(c), Associated Legendre function normalization clause (c)).

[F2]

Core action of the orbital generators. For every componentwise Schwartz class $φ \in S^{(r)}$ the orbital generators satisfy $L_{i} φ \in H_{s}$ , $S^{(r)} \subseteq D (L_{i})$ , and $L_{i} φ = - i ℏ \sum_{j} (e_{i} \times x)_{j} D_{j} φ = \sum_{j, k} ε_{i j k} Q_{j} P_{k} φ,$ with $D_{j}$ the componentwise weak derivative (Coordinate-rotation generators on the Schwartz classes and their commutators clauses (a),(b), Componentwise position and momentum operators on the component-valued wave-function space clauses (a),(c)).

[F3]

Orbital Casimir identity. For $F \in V_{l}$ one has $\sum_{i} L_{i}^{2} F = ℏ^{2} l (l + 1) F$ on the harmonic span, and the on-shell values are exactly the squared norms $\sum_{i} ∥ L_{i}^{s p h} Y_{l m} ∥^{2} = ℏ^{2} l (l + 1)$ (The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models clauses (b),(c), Orbital self-adjoint generators).

[F4]

Polar integration and the sphere measure. For a Borel $f \geq 0$ on $R^{3}$ one has $\int_{R^{3}} f d x = \int_{0}^{\infty} \int_{S^{2}} f (r Ω) r^{2} d σ (Ω) d r$ , and $\int_{S^{2}} F d σ = \int_{(0, π) \times (0, 2 π)} F \circ Φ \sin θ d (θ, φ)$ with $σ (S^{2}) = 4 π$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Sphere integrals in angle coordinates).

[F5]

Schwartz calculus and the class reading. $C_{c}^{\infty} (R^{3}; C^{r}) \subseteq S^{(r)}$ ; a classical partial derivative of a $C^{1}$ function that is locally integrable is its weak derivative; the model space consists of almost-everywhere classes (Schwartz space and its seminorms, Componentwise position and momentum operators on the component-valued wave-function space, The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions, Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument).

Proof

Proof technique: direct; the vector is a compactly supported smooth class with a homogeneous polynomial angular factor, so the core identity of the generators applies; the radial and angular integrals separate by polar integration and the Casimir value is transported from the sphere.

Given: The data of the statement and [F1]-[F5].

1.1F1F5algebra

[Smoothness and the polynomial identity] Since $supp R \subseteq [a, b]$ with $0 < a \leq b < \infty$ and $h_{l m}$ is a polynomial, the function $ψ = R (∣ x ∣) h_{l m} (x) w$ vanishes identically near $0$ and is smooth on $R^{3}$ with compact support, so $ψ \in C_{c}^{\infty} (R^{3}; C^{r}) \subseteq S^{(r)}$ by [F5]; for $x \neq 0$ and each $i$ , the product rule gives $(x \times \nabla)_{i} ψ = R (∣ x ∣) (D_{i} h_{l m}) (x) w$ because $(x \times \nabla)_{i} R (∣ x ∣) = R^{'} (∣ x ∣) (x \times x)_{i} / ∣ x ∣ = 0$ , and by [F1] the function $- i ℏ D_{i} h_{l m}$ is the polynomial representative of $L_{i}^{s p h} Y_{l m}$ ; moreover [F1] gives $D_{3} h_{l m} = i m h_{l m}$ .

2.1step 1.1F1F2F5algebra

[Core action and $L_{3}$ eigenvalue] By step 1.1 and [F2], $ψ \in D (L_{i})$ and $L_{i} ψ = \sum_{j, k} ε_{i j k} Q_{j} P_{k} ψ = - i ℏ (x \times \nabla)_{i} ψ = R (∣ x ∣) (- i ℏ D_{i} h_{l m}) (x) w$ for almost every $x$ , where the weak derivatives agree with the classical ones by [F5]; here $P_{k} = - i ℏ \partial_{k}$ on the Schwartz class, so the factor $- i ℏ$ occurs once. For $i = 3$ , [F1] gives $L_{3} ψ = - i ℏ R D_{3} h_{l m} w = ℏ m R h_{l m} w = ℏ m ψ$ , proving (a).

3.1step 2.1F1F3F4givenalgebra

[Norms and the Casimir form] Put $q_{i} : = D_{i} h_{l m}$ . Each term $x_{j} \partial_{k} h_{l m}$ is homogeneous of degree $l$ or zero, so $q_{i} (r Ω) = r^{l} q_{i} (Ω)$ for $r > 0$ ; for $l = 0$ , $q_{i} = 0$ . By step 2.1, $∥ w ∥ = 1$ and polar integration [F4], $∥ L_{i} ψ ∥^{2} = ℏ^{2} \int_{R^{3}} ∣ R (∣ x ∣) ∣^{2} ∣ q_{i} (x) ∣^{2} d x = ℏ^{2} (\int_{0}^{\infty} ∣ R (r) ∣^{2} r^{2 l + 2} d r) \int_{S^{2}} ∣ q_{i} (Ω) ∣^{2} d σ (Ω)$ . By [F1], $L_{i}^{s p h} Y_{l m} = - i ℏ q_{i} ∣_{S^{2}}$ , hence $∥ L_{i} ψ ∥^{2} = ∥ R ∥_{l}^{2} ∥ L_{i}^{s p h} Y_{l m} ∥^{2}$ . Likewise homogeneity of $h_{l m}$ and [F4] give $∥ ψ ∥^{2} = ∥ R ∥_{l}^{2} \int_{S^{2}} ∣ Y_{l m} ∣^{2} d σ = ∥ R ∥_{l}^{2}$ , using the normalization in Associated Legendre function normalization clause (d). Finally, on the finite invariant block $V_{l}$ , self-adjointness and [F3] give $\sum_{i} ∥ L_{i}^{s p h} Y_{l m} ∥^{2} = ⟨ \sum_{i} (L_{i}^{s p h})^{2} Y_{l m}, Y_{l m} ⟩ = ℏ^{2} l (l + 1)$ , so $\sum_{i} ∥ L_{i} ψ ∥^{2} = ℏ^{2} l (l + 1) ∥ ψ ∥^{2}$ , proving (b), including $l = 0$ and $R = 0$ .

4.1step 2.1step 3.1F3algebra

[Readings] Let $R \equiv̸ 0$ and put $φ : = ψ / ∥ ψ ∥$ ; by step 2.1, $L_{3} φ = ℏ m φ$ with $φ \in D (L_{3})$ , so the mean and the second moment of the law of $L_{3}$ in the pure preparation $[φ]$ are $⟨ L_{3} φ, φ ⟩ = ℏ m$ and $∥ L_{3} φ ∥^{2} = ℏ^{2} m^{2}$ , with variance $0$ ; for each $i$ , step 2.1 gives $φ \in D (L_{i})$ and step 3.1 gives $∥ L_{i} φ ∥^{2} = ∥ L_{i}^{s p h} Y_{l m} ∥^{2}$ , so the mean and second moment of the law of $L_{i}$ are finite with that second moment, and the quadratic form is $\sum_{i} ∥ L_{i} φ ∥^{2} = ℏ^{2} l (l + 1)$ , which is (c).

5.1step 1.1step 2.1step 3.1step 4.1F2∎

[Assembly and scope] Steps 1.1 and 2.1 prove (a), step 3.1 proves (b) and step 4.1 proves (c); only the spatial core identities, the sphere data and polar integration were used, no dynamics, apparatus or empirical input entered, and no operator identity for $\sum_{i} L_{i}^{2}$ on the full space was asserted, which is the scope statement (d).

Remarks

The density of the family. The vectors $R (∣ x ∣) h_{l m} (x) w$ with $R \in C_{c}^{\infty} ((0, \infty))$ span a dense subspace of the angular sector $L^{2} ((0, \infty), r^{2} d r) \otimes C Y_{l m} \otimes C w$ in the polar decomposition; this theorem uses only their eigenproperty and makes no completeness claim.
No hidden operator $(L^{2})$ . The Casimir value is stated as a quadratic-form identity. Defining a self-adjoint operator $\sum_{i} L_{i}^{2}$ on $L^{2} (R^{3})$ would require a separate closed-form and domain argument, which is not part of this page.

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Conservation of total angular momentum for a spin-independent rotation-invariant bounded-potential dynamics

Physical scope

A conditional consequence of the postulates P1, P2, P4, P5 and P6 for the declared isolated massive spin-s model with a real bounded measurable spin-independent potential invariant under every spatial rotation: the lifted coordinate rotation subgroups are one-parameter symmetry groups, and each total angular momentum component is a conserved assigned sharp observable whose outcome law is time-independent in every preparation, with conserved mean and second moment on D(L_i). The item asserts no symmetry of any concrete system, no spin-dependent or singular potential, no measurement procedure and no empirical result.

Statement

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- $s$ 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 $s$ , together with the angular momentum readings of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings, and assume the Axiom of Choice, the Axiom of Countable Choice and the Axiom of Dependent Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) and the readings of Assigned energy observable, unitary flow and the two pictures of autonomous evolution and Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables (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). Let $V : R^{3} \to R$ be real, bounded and measurable with $V \circ R^{- 1} = V$ almost everywhere for every $R \in S O (3)$ , let $H_{V} : = \frac{1}{2 m} \sum_{k = 1}^{3} P_{k}^{2} + M_{V}, D (H_{V}) = W^{2, 2} (R^{3}; C^{r}),$ be the model's spin-independent bounded-potential energy operator on $H_{s} = L^{2} (R^{3}; C^{r})$ with flow $U (t) = e^{- i t H_{V} / ℏ}$ in the reading of P4, and let $J_{i}^{t o t} = L_{i} + S_{i}$ be the total angular momenta of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings clause (d). Then:

(a) for each $i$ the lifted coordinate rotation family $t \mapsto U_{s} (A_{i} (t)) = e^{- i t J_{i}^{t o t} / ℏ}$ of The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators clause (c) is a strongly continuous one-parameter unitary group of symmetries of the declared dynamics: $U_{s} (A_{i} (t)) U (u) = U (u) U_{s} (A_{i} (t))$ for all $t, u$ and $U_{s} (A_{i} (t)) D (H_{V}) = D (H_{V})$ ;

(b) each $J_{i}^{t o t}$ is a conserved assigned sharp observable of the model with energy $H_{V}$ : for every preparation $ρ$ , every Borel $B \subseteq R$ and every $t \in R$ , $μ_{ρ (t)}^{J_{i}^{t o t}} (B) = μ_{ρ (0)}^{J_{i}^{t o t}} (B), ρ (t) = U (t) ρ (0) U (t)^{*},$ and for a unit vector $ψ \in D (L_{i}) = D (J_{i}^{t o t})$ the mean and the second moment of the law of $J_{i}^{t o t}$ in the pure preparation $[ψ (t)]$ are finite and independent of $t$ , with time-zero values $⟨ J_{i}^{t o t} ψ, ψ ⟩$ and $∥ J_{i}^{t o t} ψ ∥^{2}$ ;

(c) in the same situation each spin component $S_{i}$ and each orbital generator $L_{i}$ is a conserved assigned sharp observable in the same sense, the laws of $S_{i}$ and $L_{i}$ being time-independent in every preparation;

(d) Scope. The statement is conditional on P1, P2, P4, P5, P6 and the cited results, and concerns the declared isolated model with a real bounded measurable potential invariant under every spatial rotation. It asserts no symmetry of any concrete external field or apparatus, excludes spin-dependent, singular and unbounded potentials (which require their own domain and symmetry arguments), introduces no measurement, detector or empirical record, and makes no claim that a definite numerical value of $J_{i}^{t o t}$ is revealed at each time: the conserved object is the outcome law.

The lifted rotations use the physical-axes cover $ρ (W) = C^{- 1} ρ_{0} (W) C$ fixed in The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators; its coordinate subgroups satisfy $ρ (A_{i} (t)) = R_{i} (t)$ , so the total generators retain the stated signs and axes.

Facts & Assumptions

Given: The postulates P1, P2, P4, P5 and P6 in the stated readings; the three choice principles; the model parameters $m > 0$ , $s$ , $r = 2 s + 1$ ; the rotationally invariant real bounded measurable potential $V$ ; the energy operator $H_{V}$ and its flow $U$ ; the angular momentum operators $L_{i}$ , $S_{i}$ , $J_{i}^{t o t}$ ; a preparation $ρ$ ; a Borel set $B$ ; times $t, u$ ; a unit vector $ψ$ .

[F1]

The bounded-potential operator. $H_{V}$ is self-adjoint on $D (H_{V}) = W^{2, 2} (R^{3}; C^{r})$ as the componentwise sum of the free operator and the bounded multiplier $M_{V}$ , its domain is invariant under the flow $U (t) = e^{- i t H_{V} / ℏ}$ , which is a strongly continuous one-parameter unitary group, and $P_{k}, M_{V}$ act componentwise (Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (b),(c) applied to each component, Componentwise position and momentum operators on the component-valued wave-function space, Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b)).

[F2]

Rotation invariance of the model. For every $R \in S O (3)$ one has $U (R) H_{V} U (R)^{- 1} = H_{V_{R}}$ with $V_{R} (x) = V (R^{- 1} x)$ ; since $V_{R} = V$ almost everywhere by hypothesis, every $U (R)$ is a symmetry of $H_{V}$ in the sense of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (b), the map $R \mapsto U (R)$ is a strongly continuous unitary symmetry group, and each $L_{i}$ is conserved with time-independent law and moments in the sense of that page (Rotational and parity invariance of bounded-potential models: symmetries and conservation of angular momentum and parity laws clauses (a),(b)).

[F3]

Lifted coordinate group. For every $i$ , $t \mapsto U_{s} (A_{i} (t)) = U (R_{i} (t)) M_{s} (A_{i} (t)) = e^{- i t J_{i}^{t o t} / ℏ}$ is a strongly continuous one-parameter unitary group with $J_{i}^{t o t} = L_{i} + S_{i}$ self-adjoint on $D (L_{i})$ (The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators clauses (b),(c)).

[F4]

Componentwise commutation with internal matrices. The fibrewise matrix action $M$ of any fixed matrix commutes with the componentwise operators $P_{k}$ and $M_{V}$ and preserves their common domain $W^{2, 2} (R^{3}; C^{r})$ ; hence it commutes with $H_{V}$ on $D (H_{V})$ . If a bounded self-adjoint $S$ satisfies $S K \subseteq K S$ for a self-adjoint $K$ , then $S e^{- i t K / ℏ} = e^{- i t K / ℏ} S$ for every $t$ (The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators Facts & Assumptions (F6), Conserved spectral probability laws for commuting observables and stationary eigenrays clause (a')).

[F5]

Conservation from a symmetry group. Let $(A, u_{A})$ be an assigned sharp observable with $W (s) : = e^{- i s A / ℏ}$ for a self-adjoint $A$ . If $W (s) U (t) = U (t) W (s)$ for all $s, t$ , then the outcome law of $(A, u_{A})$ is time-independent in every preparation, and if $ψ \in D (A)$ is a unit vector the mean and second moment of the law in $[ψ (t)]$ are finite and independent of $t$ with time-zero values $⟨ A ψ, ψ ⟩$ and $∥ A ψ ∥^{2}$ (Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws clauses (a),(b), Mean, second moment and variance of the scalar spectral law, Density operators and scalar spectral probability, Trace class operator, The state space: density operators, pure and mixed states, and purity).

Proof

Proof technique: direct; combine the rotational invariance of the bounded potential with the componentwise commutation of the internal matrices, verify the symmetry property of the lifted coordinate groups, and read the conservation statement from the generator-conservation theorem.

Given: The data of the statement and [F1]-[F5].

1.1F1F4algebra

[Internal factors commute with the dynamics] Let $M$ be the fibrewise matrix action of a fixed matrix; by [F4] $M$ commutes with $P_{k}$ and with $M_{V}$ and preserves $W^{2, 2}$ , hence $M D (H_{V}) \subseteq D (H_{V})$ , $M (H_{V} ψ) = H_{V} (M ψ)$ for $ψ \in D (H_{V})$ , and $M U (t) = U (t) M$ for every $t$ : write $M = M_{1} + i M_{2}$ with $M_{1} = (M + M^{*}) / 2$ , $M_{2} = (M - M^{*}) / (2 i)$ self-adjoint fibrewise matrix actions, apply the second part of [F4] to each, and add; in particular this applies to $M = M_{s} (A_{i} (t))$ , to $M = S_{i}$ and to every bounded matrix action.

1.2F2algebra

[Spatial rotations are symmetries] By the hypothesis $V \circ R^{- 1} = V$ almost everywhere and [F2], every $U (R)$ is a symmetry of $H_{V}$ , so $U (R) U (t) = U (t) U (R)$ for all $R, t$ and $U (R) D (H_{V}) = D (H_{V})$ ; in particular this holds for the coordinate rotations $U (R_{i} (t))$ .

2.1step 1.1step 1.2F3algebra

[Lifted coordinate groups are symmetries] Fix $i$ and $t$ ; by [F3], $U_{s} (A_{i} (t)) = U (R_{i} (t)) M_{s} (A_{i} (t))$ ; by step 1.2 the first factor commutes with $U (u)$ and preserves $D (H_{V})$ , and by step 1.1 the second factor does the same, so the product commutes with $U (u)$ for every $u$ and preserves $D (H_{V})$ ; combined with the group property and strong continuity of [F3], the family $t \mapsto U_{s} (A_{i} (t))$ is a strongly continuous one-parameter unitary group of symmetries, which is (a).

2.2step 1.1F2F4F5algebra

[Spin and orbital components] By step 1.1 applied to $M = S_{i}$ , the bounded self-adjoint $S_{i}$ satisfies $S_{i} H_{V} \subseteq H_{V} S_{i}$ , so [F4] gives $S_{i} U (t) = U (t) S_{i}$ for every $t$ ; applying [F5] with $A : = S_{i}$ (bounded, so $W (s) = e^{- i s S_{i} / ℏ}$ commutes with $U$ ) gives the time-independence of the law of $S_{i}$ in every preparation, and by [F2] the law and moments of each $L_{i}$ are time-independent as well, which is (c).

3.1step 2.1F3F5algebra

[Total angular momentum conservation] By step 2.1 and [F3], the group $W (s) : = e^{- i s J_{i}^{t o t} / ℏ} = U_{s} (A_{i} (s))$ satisfies $W (s) U (t) = U (t) W (s)$ for all $s, t$ and preserves $D (H_{V})$ ; applying [F5] to the assigned sharp observable $(J_{i}^{t o t}, J s)$ of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings clause (d) gives the time-independence of its outcome law in every preparation and, for a unit $ψ \in D (L_{i}) = D (J_{i}^{t o t})$ , the conservation of the mean and second moment with the stated time-zero values, which is (b).

4.1step 2.1step 2.2step 3.1F1F5∎

[Assembly and scope] Step 2.1 proves (a), step 3.1 proves (b) and step 2.2 proves (c); only the bounded-potential domain, the almost-everywhere rotational invariance and the lifted coordinate groups were used, no spin-dependent or singular potential, apparatus or empirical input entered, and the conserved object is the outcome law, which is the scope statement (d).

Remarks

Where the spin-independence enters. Step 1.1 uses only that the dynamics is componentwise in the internal factor. A spin-dependent potential (for instance a term $S_{z} B (Q_{x})$ as on the Zeeman page) does not commute with arbitrary internal matrices, and the conservation of $J_{i}^{t o t}$ for such a dynamics requires its own analysis; the Zeeman page proves the weaker conservation of $S_{z}$ for a field along the third axis.
Orbital conservation is inherited. Clause (c) records the orbital statement of the symmetries page in the vocabulary of this page; the new content of the theorem is the lifted group of (a) and the total angular momentum of (b).

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Spin precession and the Zeeman splitting in a uniform magnetic field: Larmor angular frequency and the conservation of one component

Physical scope

A conditional consequence of the postulates P1, P2, P4, P5, P6 and of the adopted effective magnetic-moment coupling, for a uniform static magnetic field along the third axis of a declared inertial frame: self-adjointness and factorization of the Zeeman energy operator, conservation of the spin component along the field, and Larmor precession of the transverse spin expectations with angular frequency gamma_s B_z. The item concerns the declared model only; it excludes inhomogeneous and time-dependent fields, radiation and relaxation, asserts no measured precession frequency or gyromagnetic ratio, and identifies no apparatus.

Statement

(a) $H_{Z}$ is self-adjoint on $D (H_{Z})$ and $U_{Z} (t) = e^{- i t H_{0} / ℏ} e^{i t γ_{s} B_{z} S_{3} / ℏ} (t \in R),$ so in particular the transverse means satisfy, for every $t$ , $⟨ S_{1} ⟩ (t) = \cos (ω t) ⟨ S_{1} ⟩ (0) + \sin (ω t) ⟨ S_{2} ⟩ (0), ⟨ S_{2} ⟩ (t) = \cos (ω t) ⟨ S_{2} ⟩ (0) - \sin (ω t) ⟨ S_{1} ⟩ (0),$ and the transverse vector $(⟨ S_{1} ⟩ (t), ⟨ S_{2} ⟩ (t))$ is the rotation of $(⟨ S_{1} ⟩ (0), ⟨ S_{2} ⟩ (0))$ by the angle $- ω t$ in the $(1, 2)$ -plane;

(b) the component along the field is conserved: $S_{3} U_{Z} (t) = U_{Z} (t) S_{3}$ and $⟨ S_{3} ⟩ (t) = ⟨ S_{3} ⟩ (0)$ for every unit $ψ$ and every $t$ ; more generally the outcome law of $S_{3}$ is time-independent in every preparation;

(c) the conjugation identities $e^{i t H_{Z} / ℏ} S_{1} e^{- i t H_{Z} / ℏ} = \cos (ω t) S_{1} + \sin (ω t) S_{2}, e^{i t H_{Z} / ℏ} S_{2} e^{- i t H_{Z} / ℏ} = \cos (ω t) S_{2} - \sin (ω t) S_{1}$ hold as bounded operator identities, and with $S^{2} = ℏ^{2} s (s + 1) I$ (the Casimir operator of the model) also $S^{2} U_{Z} (t) = U_{Z} (t) S^{2}$ ;

(d) Scope. The statement is conditional on P1, P2, P4, P5, P6 and the adopted coupling; it concerns the uniform static field $B = (0, 0, B_{z})$ with $B_{z} \in R$ and the declared isolated model. It excludes inhomogeneous and time-dependent fields, spin-dependent spatial potentials, radiation, relaxation and environment coupling, asserts no measured precession frequency and no measured gyromagnetic parameter, and identifies no apparatus: $ω = γ_{s} B_{z}$ is a model quantity computed from assigned model parameters.

Facts & Assumptions

Given: The postulates P1, P2, P4, P5 and P6 and the magnetic-moment postulate in the stated readings; the Axiom of Choice and the Axiom of Countable Choice; the model parameters $m > 0$ , $s$ , $r = 2 s + 1$ and $γ_{s} \in R$ ; the field constant $B_{z} \in R$ ; the operators $H_{0}$ , $H_{Z}$ , $S_{i}$ , $S^{2}$ ; a unit vector $ψ$ ; real $t, θ$ .

[F1]

Free operator and componentwise structure. $H_{0}$ is self-adjoint on $D (H_{0}) = W^{2, 2} (R^{3}; C^{r})$ with flow $e^{- i t H_{0} / ℏ}$ ; its terms $P_{k}$ act componentwise and commute with every fibrewise matrix action, which preserves $W^{2, 2}$ (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b) applied componentwise, Componentwise position and momentum operators on the component-valued wave-function space clause (a), Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b)).

[F2]

Magnetic-moment postulate. With $γ_{s} \in R$ and a real bounded measurable profile $B$ , the Zeeman operator $H_{Z} = H_{0} - γ_{s} \sum_{i} S_{i} B_{i} (Q)$ with domain $D (H_{0})$ is a bounded self-adjoint perturbation of $H_{0}$ , of unit joule, and $t \mapsto e^{- i t H_{Z} / ℏ}$ is its strongly continuous unitary group in the reading of P4 (Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator clauses (i),(ii), Assigned sharp observables and the outcome law of the framework, The state space of the framework: preparations, pure rays and density operators).

[F3]

Product flow. If a bounded self-adjoint $B$ commutes with the flow $e^{i u A}$ of a self-adjoint $A$ , then $A + B$ with domain $D (A)$ is self-adjoint and $e^{- i t (A + B) / ℏ} = e^{- i t A / ℏ} e^{- i t B / ℏ}$ for all $t$ (Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow clauses (a),(c)).

[F4]

Spin algebra and the ladder conjugation. On the internal factor, $[S_{3}, S_{\pm}] = \pm ℏ S_{\pm}$ for $S_{\pm} = S_{1} \pm i S_{2}$ , $S_{1} = \frac{1}{2} (S_{+} + S_{-})$ , $S_{2} = \frac{1}{2 i} (S_{+} - S_{-})$ and $S^{2} = ℏ^{2} s (s + 1) I$ ; for the bounded exponential, $e^{i θ S_{3} / ℏ} S_{\pm} e^{- i θ S_{3} / ℏ} = e^{\pm i θ} S_{\pm}$ (Finite spin matrices clauses (a),(b),(c), The exponential series of a bounded operator, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[F5]

Conserved law from commutation. If a bounded self-adjoint $S$ satisfies $S K \subseteq K S$ for a self-adjoint $K$ , then $S e^{- i t K / ℏ} = e^{- i t K / ℏ} S$ for all $t$ (Conserved spectral probability laws for commuting observables and stationary eigenrays clause (a')); and if an assigned sharp observable $A$ satisfies $e^{- i s A / ℏ} U (t) = U (t) e^{- i s A / ℏ}$ for all $s, t$ , then its outcome law is time-independent in every preparation, with conserved mean and second moment on $D (A)$ (Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws clauses (a),(b), Mean, second moment and variance of the scalar spectral law, Density operators and scalar spectral probability, Trace class operator, The state space: density operators, pure and mixed states, and purity).

Proof

Proof technique: direct; write the Zeeman operator as a bounded perturbation commuting with the free flow, factor its evolution, conjugate the spin components by the resulting internal rotation, and read conservation of the field component from the commutation criterion.

Given: The data of the statement and [F1]-[F5].

1.1F1F2F3F4algebra

[Self-adjointness and factorization] The perturbation $B_{0} : = - γ_{s} B_{z} S_{3}$ is bounded self-adjoint by [F4]; by [F1] it commutes with every element of the free flow $e^{- i t H_{0} / ℏ}$ because it is a fibrewise matrix action commuting with the componentwise free operator; hence [F3] applies with $A : = H_{0}$ and $B : = B_{0}$ , so $H_{Z} = H_{0} + B_{0}$ is self-adjoint on $D (H_{0})$ and $e^{- i t H_{Z} / ℏ} = e^{- i t H_{0} / ℏ} e^{i t γ_{s} B_{z} S_{3} / ℏ}$ for every $t$ .

1.2F4algebra

[Conjugation of the transverse components] By [F4] the map $θ \mapsto g (θ) : = e^{i θ S_{3} / ℏ} S_{+} e^{- i θ S_{3} / ℏ}$ satisfies $g^{'} (θ) = e^{i θ S_{3} / ℏ} (i / ℏ) [S_{3}, S_{+}] e^{- i θ S_{3} / ℏ} = i g (θ)$ by the derivative formula of the bounded exponential, so $g (θ) = e^{i θ} S_{+}$ ; conjugating $S_{-}$ gives $e^{- i θ} S_{-}$ ; taking real and imaginary parts via [F4] yields $e^{i θ S_{3} / ℏ} S_{1} e^{- i θ S_{3} / ℏ} = \cos θ S_{1} - \sin θ S_{2}$ and $e^{i θ S_{3} / ℏ} S_{2} e^{- i θ S_{3} / ℏ} = \sin θ S_{1} + \cos θ S_{2}$ .

2.1step 1.1step 1.2F1algebra

[Transverse means] By step 1.1, $U_{Z} (t)^{*} = e^{- i t γ_{s} B_{z} S_{3} / ℏ} e^{i t H_{0} / ℏ}$ , so using the commutation of $S_{1}, S_{2}$ with $H_{0}$ from [F1], $U_{Z} (t)^{*} S_{1} U_{Z} (t) = e^{- i ω t S_{3} / ℏ} S_{1} e^{i ω t S_{3} / ℏ} = \cos (ω t) S_{1} + \sin (ω t) S_{2}$ and $U_{Z} (t)^{*} S_{2} U_{Z} (t) = \cos (ω t) S_{2} - \sin (ω t) S_{1}$ by step 1.2 with $θ = - ω t$ ; taking expectations in the unit vector $ψ$ gives the two displayed mean identities, and the pair transformation is the rotation by $- ω t$ ; this is (a) and the conjugation identities of (c).

2.2step 1.1F4F5algebra

[Conservation of the field component] By [F4] the bounded self-adjoint $S_{3}$ commutes with $H_{0}$ and with itself, hence $S_{3} H_{Z} \subseteq H_{Z} S_{3}$ on $D (H_{Z})$ ; [F5] gives $S_{3} U_{Z} (t) = U_{Z} (t) S_{3}$ for every $t$ , so $⟨ S_{3} ⟩ (t) = ⟨ S_{3} ⟩ (0)$ for every unit $ψ$ , and applying the second part of [F5] to the assigned sharp observable $(S_{3}, J s)$ of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings clause (c) gives the time-independence of the law of $S_{3}$ in every preparation; moreover $S^{2}$ is a scalar operator by [F4] and commutes with $U_{Z} (t)$ . This is (b) and the Casimir statement of (c).

3.1step 1.1step 2.1step 2.2F2∎

[Assembly and scope] Steps 1.1 and 2.1 prove (a) and the conjugation identities of (c), and step 2.2 proves (b) and the Casimir commutation of (c); only the uniform static profile, the componentwise free operator and the internal matrices were used, no inhomogeneous or time-dependent field, radiation, relaxation, apparatus or empirical input entered, which is the scope statement (d).

Remarks

Sign and naming of the angular frequency. The model's precession angular frequency is $ω = γ_{s} B_{z}$ , a signed quantity; the transverse vector rotates by $- ω t$ . Physical conventions differing by the sign of $γ_{s}$ or by the orientation of the field axis differ by the sign of $ω$ and by the sense of the rotation, not by the content of the theorem.
Energy eigenvalues of the internal factor. Combining this theorem with the spin spectrum page, the internal factor of the uniform-field model has the energy offsets $- γ_{s} B_{z} ℏ m$ , $m = - s, \dots, s$ , added to the free kinetic energy; the theorem does not assert a spectroscopic measurement or the absence of additional structure.

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

Physical scope

A conditional consequence of the postulates P1, P2, P4, P5, P6 and of the adopted effective magnetic-moment coupling for the spin-half model in a static field profile B = (0,0,B_z(x_1)) whose component B_z is C^1 with bounded value and derivative: the identification of the model's Zeeman operator with the smooth bounded-coefficient force model, conservation of the field-aligned spin component and of the sector weights, the conditional rate gamma_s hbar G/2 of the transverse mean momentum in the spin-up sector when the gradient equals G on the support of the transverse marginal, and the divergence-free, curl identity of the profile. The item asserts no apparatus, no measured trajectory, no beam splitting observation and no source-free field realization; it is a model prediction conditional on its stated preparation assumptions.

Statement

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 $γ_{1 / 2} \in R$ 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 ( ${A C}_{ω}$ )). Let $B_{z} \in C^{1} (R)$ be real with $B_{z}$ and $B_{z}^{'}$ bounded, let $B (x) : = (0, 0, B_{z} (x_{1})),$ let $H_{Z}$ be the Zeeman operator of the magnetic-moment postulate for this profile, acting on $H_{1 / 2} = L^{2} (R^{3}; C^{2})$ with domain $D (H_{Z}) = W^{2, 2} (R^{3}; C^{2})$ , and let $u (t) = e^{- i t H_{Z} / ℏ} u_{0}$ be its flow. Identify $H_{1 / 2}$ with $H = L^{2} (R^{3}; C) \oplus L^{2} (R^{3}; C)$ through the internal basis $(e_{1 / 2}, e_{- 1 / 2})$ , and let $P_{+} : = \frac{1}{2} (I + 2 S_{3} / ℏ)$ be the fibrewise orthogonal projection onto the spin-up line $e_{1 / 2}$ . Then:

(a) under this identification $H_{Z}$ is the operator $K (u_{+}, u_{-}) = (\frac{1}{2 m} \sum_{j} P_{j}^{2} u_{+} - \frac{γ_{1 / 2} ℏ}{2} B_{z} (Q_{x}) u_{+}, \frac{1}{2 m} \sum_{j} P_{j}^{2} u_{-} + \frac{γ_{1 / 2} ℏ}{2} B_{z} (Q_{x}) u_{-}), D (K) = W^{2, 2} \oplus W^{2, 2},$ of Conditional Stern-Gerlach force for a spin-coupled bounded field model with $γ_{s} : = γ_{1 / 2}$ ; consequently $H_{Z}$ is self-adjoint, $S_{3} H_{Z} = H_{Z} S_{3}$ on $D (H_{Z})$ , the sector weights $∥ P_{\pm} u (t) ∥^{2}$ 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 $B_{z}^{'} (x_{1}) = G$ for every $x_{1}$ in a Borel set $N \subseteq R$ satisfying $\int_{R ∖ N} η_{+} (x_{1}) d x_{1} = 0$ , where $η_{+} (x_{1}) : = \int_{R^{2}} ∣ u_{0}^{+} (x_{1}, x_{2}, x_{3}) ∣^{2} d x_{2} d x_{3}$ is the first-coordinate marginal density of the spin-up component, then $\frac{d}{d t} ∣_{t = 0} ⟨ P_{x} u (t), u (t) ⟩ = \frac{γ_{1 / 2} ℏ}{2} G;$ for $t \neq 0$ the general rate is $\frac{d}{d t} ⟨ P_{x} u (t), u (t) ⟩ = γ_{1 / 2} ⟨ S_{3} B_{z}^{'} (Q_{x}) u (t), u (t) ⟩$ ;

(c) the profile satisfies $div B = 0$ and $curl B = (0, - B_{z}^{'} (x_{1}), 0)$ at every point of $R^{3}$ , 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 $B$ nor a Maxwell solution. The word "Stern-Gerlach" names this conditional model, not a reported experiment; no measured value of $γ_{1 / 2}$ enters and no empirical conclusion is drawn.

Facts & Assumptions

Given: The postulates P1, P2, P4, P5 and P6 and the magnetic-moment postulate in the stated readings; the Axiom of Choice and the Axiom of Countable Choice; the model parameters $m > 0$ , $s = \frac{1}{2}$ , $γ_{1 / 2} \in R$ ; the field component $B_{z} \in C^{1} (R)$ with $B_{z}, B_{z}^{'}$ bounded and the profile $B = (0, 0, B_{z} (x_{1}))$ ; the operator $H_{Z}$ and the flow $u (t)$ ; a unit vector $u_{0} \in D (H_{Z}^{2})$ ; the real number $G$ .

[F1]

Model operators. On $H_{1 / 2} = L^{2} (R^{3}; C^{2})$ the componentwise operators $P_{j}, Q_{j}$ and the internal operators $S_{i} = \frac{ℏ}{2} σ_{i}$ are those of The massive spin-s particle model of the framework: state space, operators, units and frame readings clauses (b),(c) and Finite spin matrices clause (e); in the basis $(e_{1 / 2}, e_{- 1 / 2})$ one has $S_{3} (u_{+}, u_{-}) = (\frac{ℏ}{2} u_{+}, - \frac{ℏ}{2} u_{-})$ and $P_{+} = \frac{1}{2} (I + 2 S_{3} / ℏ)$ projects fibrewise onto $e_{1 / 2}$ .

[F2]

Zeeman operator of the model. For the profile $B$ with bounded measurable components the Zeeman operator of Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator clause (ii) reads $H_{Z} = H_{0} - γ_{1 / 2} S_{3} B_{z} (Q_{x})$ on $D (H_{Z}) = W^{2, 2} (R^{3}; C^{2})$ , where $B_{z} (Q_{x})$ is componentwise multiplication by $x \mapsto B_{z} (x_{1})$ ; it is self-adjoint on that domain by the postulate's bounded-perturbation statement (Assigned energy observable, unitary flow and the two pictures of autonomous evolution, The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity).

[F3]

The smooth bounded-coefficient force model. With $ℏ > 0$ , $m > 0$ , $γ_{s} \in R$ and real $B_{z} \in C^{1} (R)$ with $B_{z}$ , $B_{z}^{'}$ bounded, the operator $K (u_{+}, u_{-}) = (\frac{1}{2 m} \sum_{j} P_{j}^{2} u_{+} - \frac{γ_{s} ℏ}{2} B_{z} (Q_{x}) u_{+}, \frac{1}{2 m} \sum_{j} P_{j}^{2} u_{-} + \frac{γ_{s} ℏ}{2} B_{z} (Q_{x}) u_{-}), D (K) = W^{2, 2} \oplus W^{2, 2},$ on $H = L^{2} (R^{3}; C) \oplus L^{2} (R^{3}; C)$ is self-adjoint, $S_{z} K = K S_{z}$ on $D (K)$ for $S_{z} (u_{+}, u_{-}) = (\frac{ℏ}{2} u_{+}, - \frac{ℏ}{2} u_{-})$ , the sector weights and the mean of $S_{z}$ are conserved, for $u_{0} \in D (K^{2})$ with $∥ u_{0} ∥ = 1$ the function $t \mapsto ⟨ P_{x} u (t), u (t) ⟩$ is $C^{1}$ with $\frac{d}{d t} ⟨ P_{x} u (t), u (t) ⟩ = γ_{s} ⟨ S_{z} B_{z}^{'} (Q_{x}) u (t), u (t) ⟩$ , and if in addition $p_{+} u_{0} = u_{0}$ with $p_{\pm} : = \frac{1}{2} (I \pm \frac{2}{ℏ} S_{z})$ and $B_{z}^{'} = G$ on a Borel set carrying the first-coordinate marginal of $u_{0}^{+}$ , then $\frac{d}{d t} ∣_{t = 0} ⟨ P_{x} u (t), u (t) ⟩ = \frac{γ_{s} ℏ}{2} G$ ; furthermore $div B = 0$ and $curl B = (0, - B_{z}^{'} (x_{1}), 0)$ at every point (Conditional Stern-Gerlach force for a spin-coupled bounded field model clauses (a),(b),(c),(d), Position and momentum operators on Euclidean L2, Self-adjointness, Fourier representation and translation identities of position and momentum).

Proof

Proof technique: direct; identify the model's Zeeman operator with the smooth bounded-coefficient force model through the internal basis, transport the supplier's conservation, force and field identities, and read the physical scope.

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

1.1F1F2F3algebra

[Identification of the operators] Under the identification of $H_{1 / 2}$ with $H$ through $(e_{1 / 2}, e_{- 1 / 2})$ , the model's componentwise operators $P_{j}, Q_{j}$ of [F1] have exactly the componentwise form of the supplier's operators, whose self-adjointness and componentwise action are those of [F3]'s cited position-and-momentum items; the internal operator $S_{3}$ becomes the diagonal matrix with entries $\pm ℏ / 2$ , so $- γ_{1 / 2} S_{3} B_{z} (Q_{x}) (u_{+}, u_{-}) = (- \frac{γ_{1 / 2} ℏ}{2} B_{z} (Q_{x}) u_{+}, + \frac{γ_{1 / 2} ℏ}{2} B_{z} (Q_{x}) u_{-})$ and $H_{Z} = K$ with $γ_{s} : = γ_{1 / 2}$ , including the domain $W^{2, 2} \oplus W^{2, 2}$ ; also $P_{\pm} = p_{\pm}$ .

2.1step 1.1F3algebra

[Conservation and force] By step 1.1 and [F3], $H_{Z} = K$ is self-adjoint with $S_{3} H_{Z} = H_{Z} S_{3}$ on $D (H_{Z})$ , so the sector weights $∥ P_{\pm} u (t) ∥^{2}$ and the mean $⟨ S_{3} ⟩_{u (t)}$ are independent of $t$ ; for $u_{0} \in D (H_{Z}^{2})$ with $P_{+} u_{0} = u_{0}$ and the marginal-support hypothesis, the supplier's identities give the general rate $\frac{d}{d t} ⟨ P_{x} u (t), u (t) ⟩ = γ_{1 / 2} ⟨ S_{3} B_{z}^{'} (Q_{x}) u (t), u (t) ⟩$ and the time-zero value $\frac{γ_{1 / 2} ℏ}{2} G$ , which are (a) and (b).

2.2step 1.1F3

[Field identities] By [F3] the displayed profile satisfies $div B = 0$ and $curl B = (0, - B_{z}^{'} (x_{1}), 0)$ at every point, which is (c).

3.1step 2.1step 2.2F2∎

[Assembly and scope] Step 2.1 proves (a) and (b), and step 2.2 proves (c); only the displayed model operators, the adopted coupling and the supplier's conditional identities were used, no apparatus, trajectory, count, calibration or empirical value entered, which is the scope statement (d).

Remarks

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 $- \nabla (- μ \cdot B)$ and does not assert a classical trajectory or a definite transverse displacement of an individual particle.
The marginal hypothesis. The support condition on $η_{+}$ 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 $B_{z}^{'}$ enters pointwise.

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Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet

Physical scope

The internal-factor reading of the composition of two declared spin labels s and t under the composition postulate P1, the spin assignment P6 and the locally adopted diagonal two-factor rotation action: the total spin operators on the tensor product of the two internal factors, their commutation relations and Casimir, the Clebsch-Gordan decomposition into total-spin blocks with multiplicities, and the explicit spin-half singlet. The item concerns the internal factors only; the full two-particle spatial model, statistics, interactions and any measured correlation belong to the composites pages and are not asserted here.

Statement

Adopt the composition postulate P1 and the internal-factor assignment of the spin postulate P6 in the readings of State and composition postulate P1: Hilbert spaces, preparations and composite systems and Identical-particle and spin assignments P6: permutation sectors and finite-spin representations (The state space of the framework: preparations, pure rays and density operators, Hilbert tensor construction: inner product, completion and product bases), and assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Fix two declared spin labels $s, t \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ with internal factors $V_{s} = C^{2 s + 1}$ , $V_{t} = C^{2 t + 1}$ and spin matrices $J_{i}^{(s)}$ , $J_{i}^{(t)}$ of Finite spin matrices, and consider the composite internal factor $V_{s} \otimes V_{t}$ . On this composite, explicitly adopt the diagonal internal rotation assignment $D_{s, t} (W) : = D_{s} (W) \otimes D_{t} (W)$ for $W \in S U (2)$ , where each $D_{s}, D_{t}$ is the representation assigned by P6 clause (ii). This two-factor action is a local composite model assignment, not an additional conclusion of P6. Put $S_{i}^{t o t} : = ℏ (J_{i}^{(s)} \otimes I_{V_{t}} + I_{V_{s}} \otimes J_{i}^{(t)}), S^{t o t 2} : = \sum_{i = 1}^{3} (S_{i}^{t o t})^{2}, S_{z}^{t o t} : = S_{3}^{t o t} .$ Write $S_{i}^{(s)} : = ℏ J_{i}^{(s)}$ and $S^{(s) 2} : = \sum_{i} (S_{i}^{(s)})^{2}$ , and similarly for $t$ ; define $S_{\pm}^{t o t} : = S_{1}^{t o t} \pm i S_{2}^{t o t}$ . Then:

(a) each $S_{i}^{t o t}$ is a self-adjoint endomorphism of $V_{s} \otimes V_{t}$ satisfying $[S_{i}^{t o t}, S_{j}^{t o t}] = i ℏ \sum_{k} ε_{i j k} S_{k}^{t o t}$ , $S^{t o t 2}$ is self-adjoint and commutes with every $S_{i}^{t o t}$ , and $S^{t o t 2} = S^{(s) 2} \otimes I_{V_{t}} + I_{V_{s}} \otimes S^{(t) 2} + 2 \sum_{i = 1}^{3} S_{i}^{(s)} \otimes S_{i}^{(t)};$

(b) there is an orthogonal decomposition $V_{s} \otimes V_{t} = ⨁_{j = ∣ s - t ∣}^{s + t} W_{j}$ into $S^{t o t}$ -invariant subspaces with $\dim W_{j} = 2 j + 1$ , on each of which the triple $(S_{1}^{t o t}, S_{2}^{t o t}, S_{3}^{t o t})$ is unitarily equivalent to the spin- $j$ matrices of Finite spin matrices scaled by $ℏ$ ; in particular $S^{t o t 2} = ℏ^{2} j (j + 1) I$ on $W_{j}$ , each $j$ occurs exactly once, and there is an orthonormal basis $(v_{j, k})$ , $0 \leq k \leq 2 j$ , of $W_{j}$ with $S_{z}^{t o t} v_{j, k} = ℏ (j - k) v_{j, k}, S_{+}^{t o t} v_{j, k} = ℏ \sqrt{k (2 j - k + 1)} v_{j, k - 1}, S_{-}^{t o t} v_{j, k} = ℏ \sqrt{(k + 1) (2 j - k)} v_{j, k + 1};$ the coefficients of the $v_{j, k}$ in the product basis $(e_{m}^{(s)} \otimes e_{n}^{(t)})$ are the Clebsch-Gordan coefficients and are determined by the phase of the top vector $u_{j}$ (a unit vector of $S_{z}^{t o t}$ -weight $ℏ j$ orthogonal to all $W_{j^{'}}$ with $j^{'} > j$ ), with $u_{s + t} = e_{s}^{(s)} \otimes e_{t}^{(t)}$ admissible;

(c) for every $M$ the $S_{z}^{t o t}$ -eigenspace with weight $ℏ M$ has dimension $N (M) = # {m \in {- s, \dots, s} : M - m \in {- t, \dots, t}}$ , and the identity $(e_{m}^{(s)} \otimes e_{n}^{(t)})_{m, n}$ is an orthonormal basis of $V_{s} \otimes V_{t}$ of $S_{z}^{t o t}$ -eigenvectors with weights $ℏ (m + n)$ ;

(d) for $s = t = \frac{1}{2}$ one has $V_{1 / 2} \otimes V_{1 / 2} = W_{0} \oplus W_{1}$ with $\dim W_{0} = 1$ and $\dim W_{1} = 3$ ; the vector $Ψ : = \frac{1}{\sqrt{2}} (e_{1 / 2}^{(1 / 2)} \otimes e_{- 1 / 2}^{(1 / 2)} - e_{- 1 / 2}^{(1 / 2)} \otimes e_{1 / 2}^{(1 / 2)})$ is a unit vector with $S_{i}^{t o t} Ψ = 0$ for every $i$ , and it is the unique unit vector in the kernel of $S^{t o t 2}$ up to a phase;

(e) Scope. The statement concerns the internal factors of a declared composite, the assignments of P1 and P6 and the declared diagonal two-factor action. It asserts no spatial two-particle model, no statistics or permutation symmetry, no interaction, no dynamics, no measurement and no empirical correlation; in particular it does not by itself imply the singlet correlations of a separated pair, which belong to the composites page.

Facts & Assumptions

Given: The postulates P1 and P6 in the stated readings; the Axiom of Countable Choice; the local diagonal rotation assignment $D_{s, t}$ ; the spin labels $s, t$ ; the internal factors with their bases $(e_{m}^{(s)})$ , $(e_{n}^{(t)})$ ; the total spin operators; a real number $M$ .

[F1]

Finite spin matrices. On $V_{s}$ the matrices $J_{i}^{(s)}$ are self-adjoint with $[J_{i}^{(s)}, J_{j}^{(s)}] = i \sum_{k} ε_{i j k} J_{k}^{(s)}$ , $J^{(s) 2} = s (s + 1) I$ , $J_{+}^{(s)} e_{m}^{(s)} = \sqrt{(s - m) (s + m + 1)} e_{m + 1}^{(s)}$ and $J_{z}^{(s)} e_{m}^{(s)} = m e_{m}^{(s)}$ (and analogously for $t$ ) (Finite spin matrices clauses (a),(b),(c),(d)).

[F2]

Tensor spin composition. On $V_{s} \otimes V_{t}$ the operators $J_{i}^{t o t} : = J_{i}^{(s)} \otimes I + I \otimes J_{i}^{(t)}$ are self-adjoint with the angular momentum commutators, $J^{t o t 2} = J^{(s) 2} \otimes I + I \otimes J^{(t) 2} + 2 \sum_{i} J_{i}^{(s)} \otimes J_{i}^{(t)}$ , there is an orthogonal decomposition $V_{s} \otimes V_{t} = ⨁_{j = ∣ s - t ∣}^{s + t} W_{j}$ with $\dim W_{j} = 2 j + 1$ , each $j$ occurring once, on each block the triple is unitarily equivalent to the spin- $j$ matrices, the weight multiplicity is $N (M) = # {m \in L_{s} : M - m \in L_{t}}$ with the identity basis $(e_{m}^{(s)} \otimes e_{n}^{(t)})$ , and the block bases $v_{j, k} = (k! (2 j) \dots (2 j - k + 1))^{- 1 / 2} (J_{-}^{t o t})^{k} u_{j}$ satisfy the displayed ladder actions, with the phase of $u_{j}$ fixing the Clebsch-Gordan coefficients and $u_{s + t} = e_{s}^{(s)} \otimes e_{t}^{(t)}$ admissible (Tensor spin composition clauses (a),(b),(c),(d),(e), Hilbert tensor construction: inner product, completion and product bases).

[F3]

Internal-factor reading. P1 supplies the composite internal factor $V_{s} \otimes V_{t}$ , and P6 clause (ii) assigns $D_{s}, D_{t}$ separately. The diagonal composite action $D_{s, t} (W) = D_{s} (W) \otimes D_{t} (W)$ is adopted in the Statement. Its representation law and unitarity follow from tensor composition and adjoints, and its continuity from the finite-dimensional continuity of each factor. Along $A_{i} (u)$ its derivative at zero is $- i (J_{i}^{(s)} \otimes I + I \otimes J_{i}^{(t)})$ by the product rule and the P6 coordinate generators. Thus $S_{i}^{t o t}$ are the scaled generators of this assigned action. The model's spin observables are $S = ℏ J$ on each factor (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii), State and composition postulate P1: Hilbert spaces, preparations and composite systems, The massive spin-s particle model of the framework: state space, operators, units and frame readings, Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings).

Proof

Proof technique: direct; rescale the matrix composition theorem by $ℏ$ , transfer its decomposition and ladder statements to the total spin operators, and evaluate the spin-half singlet explicitly.

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

1.1F2algebra

[Operators and algebra] By [F2] the operators $J_{i}^{t o t}$ are self-adjoint with the angular momentum commutators and the displayed Casimir identity; multiplying by $ℏ$ and using $S_{i}^{t o t} = ℏ J_{i}^{t o t}$ , $S_{i}^{(s)} = ℏ J_{i}^{(s)}$ , $S_{i}^{(t)} = ℏ J_{i}^{(t)}$ and $S^{(s) 2} = ℏ^{2} J^{(s) 2}$ gives $[S_{i}^{t o t}, S_{j}^{t o t}] = i ℏ \sum_{k} ε_{i j k} S_{k}^{t o t}$ and the identity of (a). Each $S_{i}^{t o t}$ is self-adjoint, so its square is self-adjoint; because the space is finite-dimensional, their sum $S^{t o t 2}$ is self-adjoint. Its commutation with every $S_{i}^{t o t}$ follows from the Casimir commutation in [F2], which completes (a).

1.2F1F2algebra

[Singlet computation] Let $s = t = \frac{1}{2}$ and $Ψ = \frac{1}{\sqrt{2}} (e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)} - e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)})$ ; by [F1] $J_{+}^{(s)} e_{1 / 2}^{(s)} = 0$ , $J_{+}^{(s)} e_{- 1 / 2}^{(s)} = e_{1 / 2}^{(s)}$ and $J_{-}^{(s)} e_{1 / 2}^{(s)} = e_{- 1 / 2}^{(s)}$ , $J_{-}^{(s)} e_{- 1 / 2}^{(s)} = 0$ , and analogously for $t$ ; hence $J_{+}^{t o t} (e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)}) = e_{1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)} = J_{+}^{t o t} (e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)})$ and $J_{-}^{t o t} (e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)}) = e_{- 1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)} = J_{-}^{t o t} (e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)})$ , so $J_{\pm}^{t o t} Ψ = 0$ , giving $J_{1}^{t o t} Ψ = J_{2}^{t o t} Ψ = 0$ by $J_{\pm}^{t o t} = J_{1}^{t o t} \pm i J_{2}^{t o t}$ . Separately, [F1] gives $J_{3}^{t o t} (e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)}) = (\frac{1}{2} - \frac{1}{2}) e_{1 / 2}^{(s)} \otimes e_{- 1 / 2}^{(t)} = 0$ and $J_{3}^{t o t} (e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)}) = (- \frac{1}{2} + \frac{1}{2}) e_{- 1 / 2}^{(s)} \otimes e_{1 / 2}^{(t)} = 0$ , hence $J_{3}^{t o t} Ψ = 0$ as well; $Ψ$ is unit because the product basis is orthonormal.

2.1step 1.1F2algebra

[Decomposition and Clebsch-Gordan data] Transferring [F2] to the total spin operators by the rescaling of step 1.1 gives the orthogonal decomposition $V_{s} \otimes V_{t} = ⨁_{j = ∣ s - t ∣}^{s + t} W_{j}$ with $\dim W_{j} = 2 j + 1$ , the unitary equivalence with the spin- $j$ matrices on each block and hence $S^{t o t 2} = ℏ^{2} j (j + 1) I$ on $W_{j}$ , the weight multiplicity of (c) and the identity basis; the block bases $v_{j, k}$ have the displayed ladder actions with the stated phase convention, which are (b) and (c).

3.1step 1.2step 2.1F2algebra

[Uniqueness of the singlet] For $s = t = \frac{1}{2}$ the decomposition of step 2.1 has the two blocks $j = 0$ and $j = 1$ with dimensions $1$ and $3$ , and $S^{t o t 2} = 0$ on the one-dimensional block $W_{0}$ and $S^{t o t 2} = ℏ^{2} \cdot 2 I$ on $W_{1}$ ; hence $\ker S^{t o t 2} = W_{0}$ is one-dimensional, and since $Ψ \neq 0$ satisfies $S^{t o t 2} Ψ = 0$ by step 1.2, it spans $W_{0}$ : the kernel is the line $C Ψ$ . This is (d).

4.1step 1.1step 2.1step 3.1F3∎

[Assembly and scope] Step 1.1 proves (a), step 2.1 proves (b),(c), and step 3.1 proves (d); only the internal factors, the assignments of [F3] and the finite matrix composition theorem were used, no spatial two-particle factor, statistics, dynamics, measurement or empirical input entered, which is the scope statement (e).

Remarks

Role of the page. This theorem is the internal-factor supplier for the composites page, where the two-particle spatial factors, the preparation and the correlation analysis are added; no measured correlation or Bell statement is made here.
Phases. The block vectors are determined by the phase of the top vector; different phases multiply all Clebsch-Gordan coefficients of a block by the same phase, so all outcome probabilities built from one block are unaffected.

RemarkRemark: AI-adaptedProof: Not applicableOpen item page →

What the angular momentum and spin page adopts, proves and does not claim

Physical scope

The scope and empirical status of the angular momentum and spin page: which postulates are adopted, which conditional results are proved, which empirical parameters remain input data with uncertainty, and which laboratory, relativistic and interaction structures are excluded. The remark asserts no model fact, no measurement and no empirical conclusion.

Scope and empirical status

Adopted assumptions. This page adopts P1, P2, P4 and P5 in the readings of the framework and the spin assignment P6 of Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii) (The massive spin-s particle model of the framework: state space, operators, units and frame readings, Assigned sharp observables and the outcome law of the framework). The effective magnetic-moment coupling with its gyromagnetic parameter $γ_{s}$ and bounded field profile is the additional model assumption Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator; like the spin label, mass and charge, the parameter $γ_{s}$ is a species parameter with uncertainty and is not derived on this page.
What is proved. The local mathematics Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow and The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators supplies the bounded-perturbation flow and the lifted SU(2) rotation action with its coordinate generators with the explicit transported physical-axes cover (the literal quaternion-coordinate cover is denoted $ρ_{0}$ there), and The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models identifies the orbital Casimir on the harmonic blocks. On that basis: Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities gives the spectra and Born laws of the spin components; Orbital angular momentum eigenstates of the massive model: spherical harmonics, the l,m quantum numbers and the Casimir value describes the orbital eigenstates with the quantum numbers $l, m$ and the Casimir value; Conservation of total angular momentum for a spin-independent rotation-invariant bounded-potential dynamics proves the conservation of the total angular momentum for a spin-independent rotation-invariant bounded-potential dynamics; Spin precession and the Zeeman splitting in a uniform magnetic field: Larmor angular frequency and the conservation of one component gives the Larmor precession of the transverse spin expectations and the conservation of the field component in a uniform static field; Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet supplies the internal-factor composition with the Clebsch-Gordan decomposition and the spin-half singlet; and The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field transfers the conditional smooth-field force identity to the model.
What is not claimed. No result of this page derives any value of $γ_{s}$ , $s$ , $m$ , $g$ -factor or $ℏ$ ; no spin-orbit, fine or hyperfine structure, relativistic spinor, anomaly or radiative correction is included; no spin-statistics theorem is asserted; no atomic, molecular or solid-state model is constructed; and no statement about the compositeness or internal structure of any particle is made. The internal factor is a finite-dimensional bookkeeping device, not a rotating body.
Experiments. This page contains no experiment item: it records no apparatus, calibration, sampling, count or uncertainty. The historical Stern-Gerlach setup appears only as the name of the conditional model of The conditional Stern-Gerlach force prediction of the spin-half Zeeman model: transverse momentum rate in a smooth bounded gradient field, whose clause (b) is a prediction of the model conditional on an explicit preparation hypothesis. Identifying the operators and parameters of this page with a laboratory preparation and apparatus is a tested identification with uncertainty under Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions; the measurement statistics of sequential spin filters are treated on the measurement pages of the category, and agreement or disagreement with data there constrains the combined model and apparatus assumptions without proving them.
Consumers. Composites, Bell and identical-particle pages of the category consume the internal-factor results of Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet and the lifted rotation action of The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators; their empirical qualifications are added on those pages and are not inherited as certainty from this one.

5 · Examples, experiments and counterexamples

None yet.

Qm Angular Momentum And Spin

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow

Statement

Facts & Assumptions

Proof

Remarks

The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators

Statement

Facts & Assumptions

Proof

Remarks

The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models

Statement

Facts & Assumptions

Proof

Remarks

Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings

Definition

Remarks

Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator

Postulate

Remarks

Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities

Statement

Facts & Assumptions

Proof

Remarks

Orbital angular momentum eigenstates of the massive model: spherical harmonics, the l,m quantum numbers and the Casimir value

Statement

Facts & Assumptions

Proof

Remarks

Conservation of total angular momentum for a spin-independent rotation-invariant bounded-potential dynamics

Statement

Facts & Assumptions

Proof

Remarks

Spin precession and the Zeeman splitting in a uniform magnetic field: Larmor angular frequency and the conservation of one component

Statement

Facts & Assumptions

Proof

Remarks

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

Statement

Facts & Assumptions

Proof

Remarks

Composition of two spin labels: total spin operators, the Clebsch-Gordan block decomposition and the spin-half singlet

Statement

Facts & Assumptions

Proof

Remarks

What the angular momentum and spin page adopts, proves and does not claim

Scope and empirical status

5 · Examples, experiments and counterexamples

Sources