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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).

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources