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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.

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

Depends on

Dependency tree · 2 levels

Sources