The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions

(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.

(d) The declared two-body Coulomb model. As a further model assumption, take two distinguishable spinless massive particles with declared mass parameters $m_1,m_2>0$ (unit kg) and opposite declared charges: particle 1 carries charge $-q_{-}$ and particle 2 carries charge $+q_{+}$, where the charge magnitudes $q_{-},q_{+}>0$ are read in coulombs. Declare the pair coupling $\kappa :=\frac{q_{-}{q}_{+}}{4\pi {\epsilon }_0},{\epsilon }_0:=({\mu }_0{c}^2{)}^{-1},$ where ${\mu }_0$ is the declared magnetic constant of the SI conventions and $c$ the speed of light, both as in clause (c) of Dimensions, SI unit conventions and quantity types of the framework; the identification $\kappa =q_{-}{q}_{+}\mathrm{/}(4\pi {\epsilon }_0)$ is a constitutive assumption of the model, and $\kappa$ is read in $\mathrm{J}\mathrm{m}$. Its equal-magnitude, net-neutral submodel is $q_{-}=q_{+}=q>0$, with charges $-q,+q$ and $\kappa =q^2\mathrm{/}(4\pi {\epsilon }_0)$; this is the branch used by the later relative-dipole consumers. The two-particle state space is the Hilbert tensor product of two copies of the one-particle space of clause (a) as in Hilbert tensor construction: inner product, completion and product bases, identified with $L^2({\mathbb{R}}^6;\mathbb{C})$ through Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}); the spin-zero internal factor is trivial, and because the two particles are of different species no symmetrisation is imposed (the case of identical species belongs to the identical-particle sector of P6 and is not used here). The assigned energy operator is $H_2:=-\frac{{\mathrm{\hslash }}^2}{2m_1}{\mathrm{\Delta }}_{x_1}-\frac{{\mathrm{\hslash }}^2}{2m_2}{\mathrm{\Delta }}_{x_2}-\frac{\kappa }{\mathrm{\mid }{x}_1-x_2\mathrm{\mid }},D(H_2):=H^2({\mathbb{R}}^6;\mathbb{C}),$ which is self-adjoint by clause (c) of Two-body reduced mass. Let $\mathrm{\Phi }(x_1,x_2):=(R,r),R:=\frac{m_1{x}_1+m_2{x}_2}{M},r:=x_1-x_2,M:=m_1+m_2,$ be the centre-of-mass/relative coordinate map of that theorem, with its unitary $U_{\mathrm{\Phi }}$. The model declares the centre-of-mass/relative reading $U_{\mathrm{\Phi }}{H}_2{U}_{\mathrm{\Phi }}^{-1}=T_R\otimes I+I\otimes C_{\mu }$ of clause (c) of that theorem, with the reduced mass $\mu :=\frac{m_1{m}_2}{m_1+m_2}(\text{unit }\mathrm{k}\mathrm{g}),$ the centre-of-mass kinetic operator $T_R:=-\frac{{\mathrm{\hslash }}^2}{2M}{\mathrm{\Delta }}_R$ and the relative operator $C_{\mu }:=-\frac{{\mathrm{\hslash }}^2}{2\mu }{\mathrm{\Delta }}_r-\frac{\kappa }{\mathrm{\mid }r\mathrm{\mid }}$; the relative operator is exactly the one-centre Coulomb operator of clause (b) with mass parameter $\mu$. The energy operators $H_2$, $T_R$ and $C_{\mu }$ are each read in joules. The model further declares the fixed-nucleus reading: the one-centre model of clauses (b), (c) with mass parameter $m_1$ is read as the limiting description in which the second particle is infinitely heavy, $\mu \to m_1$ as $m_2\to \mathrm{\infty }$ at fixed $m_1,\kappa$; this is a declared limiting reading, and no convergence theorem for operators or states is asserted by it.

(e) The hydrogen-like identification. A hydrogen-like ion assignment of the two-body model is the choice $m_1=m_{\mathrm{e}}$ (electron mass parameter), $m_2=m_{\mathrm{n}}$ (nuclear mass parameter), $q_{-}=e$ (the elementary charge of clause (c) of Dimensions, SI unit conventions and quantity types of the framework) and $q_{+}=Ze$ (nuclear charge magnitude), so that $\kappa =Z{e}^2\mathrm{/}(4\pi {\epsilon }_0)$ for a declared positive integer $Z$. The hydrogen assignment is the specialization $Z=1$, $m_2=m_{\mathrm{p}}$, for which $q_{-}=q_{+}=e$ and the pair is net-neutral. These assignments use empirical mass and charge parameters; any comparison of the model's consequences with a laboratory hydrogen atom is a tested identification in the sense of clause (iii) of P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions) and carries the uncertainty of that identification. The model declares no numerical value of $m_{\mathrm{e}},m_{\mathrm{n}},e,{\mu }_0$ or $\mathrm{\hslash }$; numerical evaluations, when made, use their declared or measured values as explicitly qualified empirical inputs.

(f) Scope exclusions. Until their own items introduce them, the Coulomb and hydrogen models exclude: spin, magnetic moment couplings and spin–statistics assignments beyond the trivial spin-zero factor; relativistic corrections, in particular any fine-structure splitting of the degeneracy of the gross-structure levels; finite nuclear size, nuclear recoil beyond the two-body reduction of clause (d), and nuclear structure; QED corrections such as vacuum polarisation and the Lamb shift, and hyperfine structure; shielding and many-body modification of the $1\mathrm{/}r$ potential; the radiation extension, spontaneous emission, radiation reaction and any photon, transition-frequency or line-intensity statement; external electromagnetic fields; and any scattering, ionisation or continuum process, which belongs to the scattering items of the quantum-mechanics track. The model asserts no empirical result and fixes no measured quantity.