A radial bounded potential: rotational and parity symmetry and conservation of the rotation generators

Adopt the postulates P1, P2, P4, P5 and P6 in the readings of 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, 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 for a declared particle of mass parameter $m>0$ and spin label $s$, with state space $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, $r=2s+1$, componentwise operators $Q_j,P_k$, rotation action $U(R)$ and coordinate-rotation generators $L_i$ (The spatial rotation action on the component-valued wave-function space, Coordinate-rotation generators on the Schwartz classes and their commutators); 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 (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Let $v:[0,\mathrm{\infty })\to \mathbb{R}$ be bounded and measurable, and let $V:{\mathbb{R}}^3\to \mathbb{R}$ be real, bounded and measurable with $V(x)=v(\mathrm{\mid }x\mathrm{\mid })\text{for almost every }x\in {\mathbb{R}}^3,$ a radial potential almost everywhere (if $V$ is the composition $x\mapsto v(\mathrm{\mid }x\mathrm{\mid })$ everywhere, this holds with empty exceptional set). Let $H_V=\frac{1}{2m}{\sum }_{k=1}^3{P}_k^2+M_V,D(H_V)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r),$ be the bounded-potential energy operator. Then:

(a) for every $R\in SO(3)$ one has $V(R^{-1}x)=V(x)$ for almost every $x$, and $V(-x)=V(x)$ for almost every $x$;

(b) every $U(R)$ is a symmetry of $H_V$, the map $R\mapsto U(R)$ is a strongly continuous unitary symmetry group of the declared dynamics, and each generator $L_i$ is a conserved observable: for every preparation $\rho$, Borel $B\subseteq \mathbb{R}$ and $t$, ${\mu }_{\rho (t)}^{L_i}(B)={\mu }_{\rho (0)}^{L_i}(B),$ with conserved mean and second moment for unit vectors in $D(L_i)$ and time-zero values $⟨L_i\psi ,\psi ⟩$, $\mathrm{\parallel }{L}_i\psi {\mathrm{\parallel }}^2$, in the unit $\mathrm{J}\mathrm{s}$ of the model's action (Dimensions, SI unit conventions and quantity types of the framework);

(c) the parity action $\mathrm{\Pi }$ is a symmetry of $H_V$, and its outcome law in every preparation is time-independent;

(d) Scope. The example is conditional on the postulates and the bounded radial class; it computes no eigenvalue, no bound state, no degeneracy and no radial solution, and the boundedness of $v$ excludes the Coulomb potential and other unbounded radial models, which require their own self-adjointness and spectral analysis. No claim is made that any particular physical potential is radial, and no empirical result is recorded.

• What makes the example work. Only two ingredients are used: the radial identity $V(x)=v(\mathrm{\mid }x\mathrm{\mid })$ almost everywhere and the orthogonality of rotations through $\mathrm{\mid }{R}^{-1}x\mathrm{\mid }=\mathrm{\mid }x\mathrm{\mid }$; the boundedness of $v$ keeps the model inside the bounded-potential class of the page, whose self-adjointness and conservation results are already established.
• What an angular-momentum conservation law is here. The conserved object is the outcome law of the assigned generator observable $L_i$ (with its moments where they exist); no classical angular-momentum vector, no definite value and no measured spectral line is asserted.