The declared central-potential model: spherical reduction of smooth compactly supported radial states

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the kinematics and massive-particle model postulate P5, read through the declared central-potential model of The declared central-potential model: state space, assigned energy operator, spherical reading and bound-state vocabulary with mass parameter $m>0$, spin label $s=0$, a real bounded measurable central potential $V(x)=v(\mathrm{\mid }x\mathrm{\mid })$ and assigned energy operator $H_V$ (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, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Let $l\ge 0$, $-l\le \bar{m}\le l$, let $u\in C_c^{\mathrm{\infty }}((0,\mathrm{\infty });\mathbb{C})$, and define $\psi (x):=\frac{u(r)}{r}{Y}_{l\bar{m}}(\mathrm{\Omega })(x=r\mathrm{\Omega },r=\mathrm{\mid }x\mathrm{\mid }>0),$ with $(Y_{lm})$ the complete orthonormal family and $U$ the unitary of Angular radial unitary reduction. Then:

(a) $\psi$ lies in $C_c^{\mathrm{\infty }}({\mathbb{R}}^3\setminus \{0\})\subseteq H^2({\mathbb{R}}^3;\mathbb{C})=D(H_V)$, its image under $U$ is the finite-support vector whose only potentially nonzero entry is the $(l,\bar{m})$-entry $u$, and $B\mapsto {\int }_B\mathrm{\mid }\psi {\mathrm{\mid }}^{2}dx$ is its position quadratic measure. For $u\equiv ̸0$, the unit vector $\widehat{\psi }:=\psi \mathrm{/}\mathrm{\parallel }\psi \mathrm{\parallel }$ represents its pure preparation and has position outcome law ${\mu }_{\widehat{\psi }}^G(B)=\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-2}{\int }_B\mathrm{\mid }\psi {\mathrm{\mid }}^{2}dx;$

(b) for almost every $x\in {\mathbb{R}}^3\setminus \{0\}$, $H_V\psi (x)=\frac{1}{r}[-\frac{{\mathrm{\hslash }}^2}{2m}{u}^{\mathrm{\prime }\mathrm{\prime }}(r)+(v(r)+\frac{{\mathrm{\hslash }}^{2}l(l+1)}{2m{r}^2})u(r)]Y_{l\bar{m}}(\mathrm{\Omega }),$ the two sides being equal as $L^2({\mathbb{R}}^3;\mathbb{C})$ classes; thus the model's assigned energy acts on the smooth spherical sector by the declared radial differential expression with effective potential $v_l^{\mathrm{e}\mathrm{f}\mathrm{f}}(r)=v(r)+\frac{{\mathrm{\hslash }}^{2}l(l+1)}{2m{r}^2}$;

(c) if $E\in \mathbb{R}$ and $u$ satisfies the declared radial equation $-\frac{{\mathrm{\hslash }}^2}{2m}{u}^{\mathrm{\prime }\mathrm{\prime }}(r)+(v(r)+\frac{{\mathrm{\hslash }}^{2}l(l+1)}{2m{r}^2})u(r)=Eu(r)(0<r<\mathrm{\infty })$ pointwise on $(0,\mathrm{\infty })$, then $H_V\psi =E\psi$, when $u\equiv ̸0$, $\psi$ is a nonzero stationary vector of the model with energy $E$ and the pure preparation $[\psi ]$ is stationary, every assigned sharp observable having a time-independent outcome law in it. For $u=0$, only the linear identity $H_{V}0=E0$ is asserted and no pure preparation is assigned;

(d) for every $u\in C_c^{\mathrm{\infty }}((0,\mathrm{\infty });\mathbb{C})$, including $u=0$, the radial equation $-\frac{{\mathrm{\hslash }}^2}{2m}{u}^{\mathrm{\prime }\mathrm{\prime }}(r)+(v(r)+\frac{{\mathrm{\hslash }}^{2}l(l+1)}{2m{r}^2})u(r)=Eu(r)$ holds for almost every $r>0$ if and only if $H_V\psi =E\psi$ as Hilbert-space vectors. If $v$ is continuous on $(0,\mathrm{\infty })$, this radial equality is also equivalent to its pointwise version. For general measurable $v$, no pointwise converse is asserted.

The statement is conditional on P1, P2, P4, P5 and the declared model; it asserts no converse for general reduced functions, no completeness of any family of radial solutions, no discrete spectrum, and no measured energy, and it does not apply to the Coulomb potential, whose self-adjointness and spectral theory are supplied separately.

• Only the sector $(l,\bar{m})$. The statement concerns the one-parameter family of smooth vectors whose reduced function is supported away from the origin in the single sector $(l,\bar{m})$; no assertion is made about a general element of $D(H_V)$, for which the spherical components are only $L^2$ classes.
• Exact scope of the converse. Clause (d) concerns the specified smooth compactly supported reduced functions and equality almost everywhere for measurable potentials. Changing $v$ at a single radius leaves its multiplication operator unchanged and cannot be detected by the Hilbert-space equation, so that equation cannot force a pointwise radial equality for an arbitrary measurable representative. A converse for arbitrary reduced classes would additionally require half-line regularity and endpoint theory; the Coulomb items supply their own separate realization.
• Units. The radial expression carries the dimension of an energy by the conventions of Dimensions, SI unit conventions and quantity types of the framework: ${\mathrm{\hslash }}^2\mathrm{/}(2m)$ times a second derivative in $r$ has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$, and the barrier term ${\mathrm{\hslash }}^{2}l(l+1)\mathrm{/}(2m{r}^2)$ likewise.