Qm Central Potentials And Hydrogen

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the choice hypothesis used by the change-of-variables lemma cited below. For every real $\lambda >0$ and every integer $k\ge 0$ the function $r\mapsto r^k{e}^{-\lambda r}$ is Lebesgue integrable on $(0,\mathrm{\infty })$ and

${\int }_0^{\mathrm{\infty }}{r}^k{e}^{-\lambda r}dr=\frac{k!}{{\lambda }^{k+1}}\mathrm{.}$

The assertion is an identity for the Lebesgue integral of a nonnegative Borel function; no probabilistic interpretation and no normalised density are asserted. In particular no statement is made about any other exponent, about complex $\lambda$, or about negative powers.

• Only the half-line is used. The change of variables is applied to the open half-line $(0,\mathrm{\infty })$, which is an open subset of $\mathbb{R}$; no endpoint value of the integrand at $0$ or at $+\mathrm{\infty }$ is used, and the equality is an equality in $[0,\mathrm{\infty }]$.
• Scaling convention. The exponent $k+1$ in the denominator and the factorial $k!$ in the numerator are fixed by the convention that the unscaled integral is $\mathrm{\Gamma }(k+1)=k!$; in particular the mean of the normalised density $r\mapsto \lambda e^{-\lambda r}$ is $1\mathrm{/}\lambda$.

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 $m,\mathrm{\hslash }>0$, put $c:=2{\pi }^2{\mathrm{\hslash }}^2\mathrm{/}m$, and let $H_0$ be the free operator of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b) with $d=3$, this $c$ and $b=0$: $H_0:={\mathcal{F}}_2^{\ast }{M}_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathcal{F}}_2,D(H_0):=\{f\in L^2({\mathbb{R}}^3;\mathbb{C}):\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f\in L^2({\mathbb{R}}^3;\mathbb{C})\},$ where ${\mathcal{F}}_2$ is the unitary Plancherel transform of that theorem. Let $D_j^2$ denote the second weak partial derivative in the $j$-th coordinate, defined componentwise on the locally integrable classes of ${\mathbb{R}}^3$ (Weak derivative of a locally integrable function), write ${\mathrm{\Delta }}_w:=D_1^2+D_2^2+D_3^2$ for the weak Laplacian (defined on the classes whose three second weak partial derivatives lie in $L^2$), and let $W^{2,2}({\mathbb{R}}^3;\mathbb{C})$ be the Sobolev space (Integer-order Sobolev spaces and their norms). Then:

(a) $D(H_0)=W^{2,2}({\mathbb{R}}^3;\mathbb{C})$ as sets and $H_{0}f=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_{w}f\text{for every }f\in D(H_0);$ equivalently, ${\mathcal{F}}_2(H_{0}f)(\xi )=c\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f(\xi )$ and ${\mathcal{F}}_2(-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_{w}f)(\xi )=c\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f(\xi )$ for almost every $\xi$, and the two $L^2$ classes agree.

(b) For every real bounded measurable $V:{\mathbb{R}}^3\to \mathbb{R}$, the operator $H_0+M_V$ of clause (c) of the cited theorem, with domain $D(H_0)$, coincides with the operator $f\mapsto -\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_{w}f+Vf$ on the same domain; in particular, for $E\in \mathbb{R}$ and $\psi \in W^{2,2}({\mathbb{R}}^3;\mathbb{C})$ the eigenvalue equation $(H_0+M_V)\psi =E\psi$ is equivalent to the weak equation $-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_w\psi +V\psi =E\psi$ elementwise in $L^2({\mathbb{R}}^3;\mathbb{C})$.

The statement identifies operators on the same domain; it asserts neither smoothness of the classes in $W^{2,2}$ nor any physical interpretation.

• Why the factor $c$. With the Plancherel convention $\widehat{f}(\xi )=\int f(x)e^{-2\pi ix\cdot \xi }dx$ the weak Laplacian has multiplier $-4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2$, so $-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_w$ has multiplier $\frac{2{\pi }^2{\mathrm{\hslash }}^2}{m}\mathrm{\mid }\xi {\mathrm{\mid }}^2$: the constant $c$ of the cited theorem is fixed by this convention and is not a convention-free physical statement.
• Scope. The lemma compares two mathematical operators on $L^2({\mathbb{R}}^3;\mathbb{C})$. It asserts no smoothness of $W^{2,2}$ classes, no boundary values, no self-adjointness beyond that of the cited theorem, and no identification of $H_0$ with any measured Hamiltonian.

(a) Framework data. 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, the identical-particle and spin postulate P6, and the preparation and apparatus discipline P7 (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, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Fix a mass parameter $m>0$ and a spin label $s=0$, so that the framework's one-particle model The massive spin-s particle model of the framework: state space, operators, units and frame readings reads its state space as $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$ with the pairing linear in the first argument, its componentwise position family $Q_1,Q_2,Q_3$, its componentwise momentum family $P_1,P_2,P_3$ and the action constant $\mathrm{\hslash }>0$; the spin-zero internal factor carries the trivial representation of P6. Units are those of Dimensions, SI unit conventions and quantity types of the framework: position is read in metres, momentum in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, energy and potential energy in joules, and $\mathrm{\hslash }$ in $\mathrm{J}\mathrm{s}$. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) The declared central-potential model. As an additional model assumption of the pair, adopt the central-potential model. Its declared state space is $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$ of clause (a), with the position and momentum families $Q_j$, $P_k$ of that clause (The massive spin-s particle model of the framework: state space, operators, units and frame readings). A declared central potential of the model is a function $V:{\mathbb{R}}^3\to \mathbb{R}$ of the form $V(x):=v(\mathrm{\mid }x\mathrm{\mid })$ for $x\mathrm{\ne }0$, where $v:(0,\mathrm{\infty })\to \mathbb{R}$ is real, bounded and measurable and $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}:=\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}={\sup }_{r>0}\mathrm{\mid }v(r)\mathrm{\mid }$ is finite; the assignment $V(0):=0$ is a declared convention that does not affect the $L^2$ theory. The potential energy is read in joules. The model declares the Born position law of the framework for a unit class $\psi$, ${\mu }_{\psi }^G(B):={\int }_B\mathrm{\mid }\psi (x){\mathrm{\mid }}^{2}dx(B\subseteq {\mathbb{R}}^3\text{ Borel}),$ the spin-zero reading of clause (b) of Position and momentum outcome laws of the massive spin-s model. The pair asserts no derivation of a central force law from $H_0$ or from P5: the restriction to a declared central potential is an assumption of the model, and an assignment of a concrete material system to the model is an identification in the sense of clause (iii) of P7 and carries the corresponding uncertainty.

(c) The assigned energy operator. Let ${\mathcal{F}}_2$ be the unitary Plancherel transform of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b), $c:=2{\pi }^2{\mathrm{\hslash }}^2\mathrm{/}m$, and let $H_0^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}:={\mathcal{F}}_2^{\ast }{M}_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathcal{F}}_2,D(H_0^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}):=\{f\in H_0:\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f\in H_0\}$ be the free operator of that clause with $d=3$ and $b=0$. With $M_V$ the bounded multiplication operator of clause (a) of the same theorem, the assigned energy operator of the model with central potential $V$ is $H_V:=H_0^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}+M_V,D(H_V):=D(H_0^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}})\mathrm{.}$ The declared energy unit is the joule. By clause (c) of the cited theorem $H_V$ is self-adjoint on $D(H_V)$, bounded below by $-\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$; by The three-dimensional free Schrödinger operator as the weak Laplacian on its Sobolev domain the domain is the Sobolev space $W^{2,2}({\mathbb{R}}^3;\mathbb{C})$ and $H_{V}f=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_{w}f+Vf(f\in D(H_V)),$ ${\mathrm{\Delta }}_w=D_1^2+D_2^2+D_3^2$ the weak Laplacian of that lemma. If $\psi \in D(H_V)\setminus \{0\}$ and $E\in \mathbb{R}$ satisfy $H_V\psi =E\psi$, the time-independent equation $-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_w\psi +V\psi =E\psi$ is an equality in $L^2({\mathbb{R}}^3;\mathbb{C})$ and hence in distributions; it is a partial differential equation in the three spatial variables. If $V\in C^2(\mathrm{\Omega })$ on an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$, then the class $\psi$ has a representative in $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,\alpha }(\mathrm{\Omega })$ for every $0<\alpha <1\mathrm{/}2$, and the equation holds pointwise there. To justify this, write $f:=(2m\mathrm{/}{\mathrm{\hslash }}^2)(E-V)\psi$, so $-\mathrm{\Delta }\psi =f$ in distributions. On every relatively compact open subset of $\mathrm{\Omega }$, the weak product rule (obtained by local smooth mollification of the $C^2$ factor and passage to the $L^2$ limit) gives, for every multi-index $\beta$ with $\mathrm{\mid }\beta \mathrm{\mid }\le 2$, $D^{\beta }f=\frac{2m}{{\mathrm{\hslash }}^2}{\sum }_{\eta \le \beta }\left(\genfrac{}{}{0px}{}{\beta }{\eta }\right)D^{\eta }(E-V)D^{\beta -\eta }\psi \in L^2;$ the factors $D^{\eta }(E-V)$ are bounded on compact subsets and $\psi \in W^{2,2}({\mathbb{R}}^3)$, so $f\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$. Integration by parts against compactly supported smooth test functions makes $\psi \in H^1(\mathrm{\Omega })$ a local weak solution in the sense of Local weak solutions of a divergence-form operator for $L=-D_i({\delta }_{ij}{D}_j)=-\mathrm{\Delta }$, whose divergence coefficients are $a^{ij}={\delta }_{ij}$ and $b^i=c=0$. They satisfy the uniform ellipticity condition of Uniformly elliptic divergence-form operators and their sesquilinear forms with $\theta =1$ and have the required local coefficient derivatives of every order. The interior regularity theorem Interior $H^{k+2}$ elliptic regularity with $k=2$ therefore gives $\psi \in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^4(\mathrm{\Omega })$. Every ball compactly contained in $\mathrm{\Omega }$ is a bounded smooth domain and hence a $W^{4,2}$ extension domain by Bounded C^k domains admit integer-order Sobolev extension. The higher-order Sobolev embedding Higher-order Sobolev embedding with $n=3$, $k=4$, $p=2$ then gives a representative with continuous derivatives through order two and locally Hölder-continuous second derivatives of every exponent $0<\alpha <1\mathrm{/}2$; this is the meaning of $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,\alpha }(\mathrm{\Omega })$ here. These representatives agree on overlaps because they are continuous and represent the same almost-everywhere class. Both sides of the equation are then continuous and agree almost everywhere, so they agree pointwise. The model adopts $(H_V,\mathrm{J})$ as its declared energy observable; its self-adjointness is the cited theorem's, and no further spectral property is presupposed here.

(d) The declared spherical reading. Let $U$ be the unitary of Angular radial unitary reduction from $H_0$ onto $K={⨁}_{(l,\bar{m})}{L}^2((0,\mathrm{\infty }),dr;\mathbb{C})$, written $U\psi =(u_{l\bar{m}}{)}_{l\ge 0,-l\le \bar{m}\le l}$, and let $(Y_{l\bar{m}})$ be the complete orthonormal family of that theorem; $l,\bar{m}$ are integers, while $m>0$ continues to denote the mass. For a fixed $(l,\bar{m})$, let ${\iota }_{l\bar{m}}:L^2((0,\mathrm{\infty }),dr;\mathbb{C})\to K$ insert its argument in that coordinate and zero in every other coordinate, and define the isometric spherical lift $J_{l\bar{m}}:=U^{-1}{\iota }_{l\bar{m}}$, so that $J_{l\bar{m}}u(x)=\frac{u(r)}{r}{Y}_{l\bar{m}}(\mathrm{\Omega })(x=r\mathrm{\Omega },r>0)\mathrm{.}$ The spherical sector $(l,\bar{m})$ is the range of this lift. Its admissible reduced domain is fixed by the assigned energy domain: ${\mathcal{D}}_{l\bar{m}}:=\{u\in L^2((0,\mathrm{\infty }),dr;\mathbb{C}):J_{l\bar{m}}u\in W^{2,2}({\mathbb{R}}^3;\mathbb{C})\}\mathrm{.}$ Thus every weak Cartesian derivative through order two of the lift must belong to $L^2$ on all of ${\mathbb{R}}^3$, including neighbourhoods of the origin. This condition and the global square integrability specify admissibility at both endpoints; no independent half-line boundary parameter is part of the model. In particular $C_c^{\mathrm{\infty }}((0,\mathrm{\infty });\mathbb{C})\subseteq {\mathcal{D}}_{l\bar{m}}$ by clause (d) of the cited theorem. The effective potential is $v_l^{\mathrm{e}\mathrm{f}\mathrm{f}}(r):=v(r)+\frac{{\mathrm{\hslash }}^{2}l(l+1)}{2m{r}^2}(r>0),$ whose second summand is called the angular-momentum barrier; both summands, and $v_l^{\mathrm{e}\mathrm{f}\mathrm{f}}$, are read in joules. For $E\in \mathbb{R}$, the model's declared radial equation in the sector $(l,\bar{m})$ is $u\in {\mathcal{D}}_{l\bar{m}},H_V{J}_{l\bar{m}}u=E{J}_{l\bar{m}}u\text{in }{L}^2({\mathbb{R}}^3;\mathbb{C})\mathrm{.}$ A radial eigenpair additionally requires $u\mathrm{\ne }0$. The associated one-dimensional differential expression is $u⟼-\frac{{\mathrm{\hslash }}^2}{2m}{u}^{\mathrm{\prime }\mathrm{\prime }}+(v(r)+\frac{{\mathrm{\hslash }}^{2}l(l+1)}{2m{r}^2})u,u:(0,\mathrm{\infty })\to \mathbb{C},$ where $u^{\mathrm{\prime }\mathrm{\prime }}$ means the distributional second derivative on $(0,\mathrm{\infty })$ (Weak derivative of a locally integrable function). The expression alone, without the specified lifted domain, does not define a radial eigenvalue problem. On $C_c^{\mathrm{\infty }}((0,\mathrm{\infty });\mathbb{C})$, its equation with right-hand side $Eu$ is equivalent almost everywhere to the declared radial equation above by the spherical-reduction theorem of this page; pointwise equivalence additionally requires continuity of $v$. No full-domain differential reduction or endpoint classification is presupposed here. The angular-momentum quantum numbers $l$ and $\bar{m}$ appearing in the reading are labels of the orthonormal family $(Y_{l\bar{m}})$; the identification of $l$ with a total angular momentum of a material system is a separate assignment of the framework's angular-momentum items and is not made here.

(e) Stationary and bound-state vocabulary. A stationary vector of the model with potential $V$ and energy $E\in \mathbb{R}$ is a nonzero class $\psi \in D(H_V)$ with $H_V\psi =E\psi$; the pair $(\psi ,E)$ is an eigenpair of the model. A bound state of the model is a stationary vector with energy $E<0$ whose representative is normalised, $\mathrm{\parallel }\psi \mathrm{\parallel }=1$; its eigenvalue is its binding energy, read in joules, relative to the chosen energy reference $0$. Here $E<0$ is a vocabulary convention; no assertion that $0$ is a spectral continuum threshold follows for an arbitrary bounded central potential. A continuum threshold requires a separate spectral determination for the particular operator. In particular, adding a constant potential shifts the free operator and its spectrum by that constant without changing its eigenvectors. The model reads the energy outcome law of the pure preparation of a unit eigenvector with eigenvalue $E$ as the point mass ${\delta }_E$, the reading of clause (c) of Assigned sharp observables and the outcome law of the framework, and the position density of that preparation as $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ by Position and momentum outcome laws of the massive spin-s model clause (b), with $s=0$ and the state, position and choice hypotheses of (a). These are declarations of the model under P2; they construct no sample, detector or measured value, and they assert no stability, decay or formation claim.

(f) Status and scope. Clauses (a)-(e) fix the data of the pair: a declared state space, position and momentum assignments, an assigned energy operator for a real bounded measurable central potential, a declared spherical reading with its effective potential and radial equation, and a stationary-state and bound-state vocabulary. They adopt P1, P2, P4, P5, P6, P7, the choice hypotheses of the cited mathematics, and the additional central-potential model assumption and spherical reading of clauses (b) and (d); in particular the central form of the potential, the model's use of the full three-dimensional space as its state space and the identification of the spherical sectors with angular-momentum labels are declarations of the pair. The model excludes from its scope, until their own items introduce them: the Coulomb potential $\propto -1\mathrm{/}r$ and other unbounded or singular potentials; spin and magnetic couplings; relativistic corrections; finite nuclear size; the radiation extension, spontaneous emission and radiation reaction; external electromagnetic fields; and any many-body or shielding modification of a central potential. No numerical value of $m$, $\mathrm{\hslash }$ or of any potential is fixed here, and no empirical result is asserted.

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.

(a) Framework data. Adopt the framework data of clause (a) of the declared central-potential model The declared central-potential model: state space, assigned energy operator, spherical reading and bound-state vocabulary with a declared mass parameter $m>0$, spin label $s=0$, the state space $H_0=L^2({\mathbb{R}}^3;\mathbb{C})$, the componentwise position and momentum families and the action constant $\mathrm{\hslash }>0$ of The massive spin-s particle model of the framework: state space, operators, units and frame readings; units are those of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) The declared one-centre Coulomb model. As an additional model assumption of the pair, fix a real coupling $\kappa >0$ with the declared unit $\mathrm{J}\mathrm{m}$ and put $\gamma :=\frac{2m\kappa }{{\mathrm{\hslash }}^2}(\text{unit }{\mathrm{m}}^{-1}),C_{\gamma }:=-\mathrm{\Delta }-\frac{\gamma }{\mathrm{\mid }\cdot \mathrm{\mid }}\text{ on }D(C_{\gamma })=H^2({\mathbb{R}}^3;\mathbb{C}),$ the self-adjoint operator of Coulomb infinitesimal kinetic bound clause (c). The assigned energy operator of the one-centre Coulomb model is $H_{\kappa }:=\frac{{\mathrm{\hslash }}^2}{2m}{C}_{\gamma },$ with domain $D(H_{\kappa }):=D(C_{\gamma })$; by Positive scalar multiples of self-adjoint operators: domains, resolvents, spectra and eigenbases clauses (a) and (b) it is self-adjoint with the same domain, and by The three-dimensional free Schrödinger operator as the weak Laplacian on its Sobolev domain together with the action formula of the cited theorem it acts by $H_{\kappa }f=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_{w}f-\kappa \frac{f}{\mathrm{\mid }\cdot \mathrm{\mid }}(f\in D(H_{\kappa })),$ where ${\mathrm{\Delta }}_w$ is the weak Laplacian. The model declares the length scale $a_{\kappa }:={\mathrm{\hslash }}^2\mathrm{/}(m\kappa )$, read in metres, and the energy scale ${\mathcal{E}}_{\kappa }:=m{\kappa }^2\mathrm{/}(2{\mathrm{\hslash }}^2)={\mathrm{\hslash }}^2\mathrm{/}(2m{a}_{\kappa }^2)$, read in joules. The model adopts $(H_{\kappa },\mathrm{J})$ as its declared energy observable for the one-centre problem; the self-adjointness is the cited theorem's and no spectral property is presupposed here.

(c) The declared radial reading of the one-centre model. Let $U$ and $(A_l{)}_{l\ge 0}$ be the unitary and the radial operators of Transported radial endpoint realization clauses (a)-(d): $U$ maps $H_0$ onto $K={⨁}_{(l,m)}{L}^2((0,\mathrm{\infty }),dr;\mathbb{C})$ with $U{H}_{\kappa }{U}^{\ast }={⨁}_{(l,m)}({\mathrm{\hslash }}^2\mathrm{/}2m)A_l$, and each $A_l$ is self-adjoint with form domain $V_l$ and domain $D(A_l)$. The model declares the sector $(l,m)$ radial energy operator $A_l^{(\kappa )}:=({\mathrm{\hslash }}^2\mathrm{/}2m)A_l$, read in joules, and, for a class $u\in D(A_l)$, the declared radial equation is the distributional identity $-u^{\mathrm{\prime }\mathrm{\prime }}(r)+(\frac{l(l+1)}{r^2}-\frac{\gamma }{r})u(r)=A_{l}u(r)\text{on }(0,\mathrm{\infty }),$ which by clause (c) of the cited theorem is an ordinary differential equation with $C^2$ representative whenever $A_{l}u=\lambda u$ for a constant $\lambda$ (unit ${\mathrm{m}}^{-2}$); its physical energy is $E=({\mathrm{\hslash }}^2\mathrm{/}2m)\lambda$ in joules. For an inverse-length parameter $k>0$ (unit ${\mathrm{m}}^{-1}$), put $\lambda =-k^2$ and $E(k):=-{\mathrm{\hslash }}^2{k}^2\mathrm{/}(2m)$ (unit $\mathrm{J}$). The regular reduced solution at this parameter is the function $u_{\mathrm{r}\mathrm{e}\mathrm{g}}(r)=r^{l+1}{e}^{-kr}v(2kr)$ of clause (c) of Kummer recurrence and termination, with $v$ the regular power series of clause (b) and dimensionless argument $2kr$; the pair uses it as the declared radial template of the model, not as a physical state, until the normalisation and spectral theorems of the page identify it.