The one-centre Coulomb model: position and radial distributions of the bound states, with the ground-state density and its scales

Adopt the framework postulates P1, P2, P4 and P5, the readings of the declared one-centre Coulomb model The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clauses (a)-(c) with parameters $\mathrm{\hslash }>0$, $m>0$, $\kappa >0$ and length scale $a_{\kappa }={\mathrm{\hslash }}^2\mathrm{/}(m\kappa )$, the energy-observable reading of The one-centre Coulomb model: energy observable, flow, stationary states and spectral support, the bound-state data of The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum, and the spin-zero position law Position and momentum outcome laws of the massive spin-s model (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). 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). For the integer labels $n\ge 1$, $0\le l\le n-1$, $-l\le \bar{m}\le l$ with magnetic label $\bar{m}$ distinct from mass $m$, let ${\psi }_{nl\bar{m}}$ and ${\mathrm{\Psi }}_{nl\bar{m}}={\psi }_{nl\bar{m}}\mathrm{/}\mathrm{\parallel }{\psi }_{nl\bar{m}}\mathrm{\parallel }$ be the bound vectors of the cited theorem, and write $u_{nl}$ for the reduced regular solution with ${\psi }_{nl\bar{m}}(x)=u_{nl}(r)Y_{l\bar{m}}(\mathrm{\Omega })\mathrm{/}r$. Then:

(a) for every label and every Borel $B\subseteq {\mathbb{R}}^3$ the position outcome law in the unit preparation $[{\mathrm{\Psi }}_{nl\bar{m}}]$ is ${\mu }_{{\mathrm{\Psi }}_{nl\bar{m}}}^G(B)=\mathrm{\parallel }{\psi }_{nl\bar{m}}{\mathrm{\parallel }}^{-2}{\int }_0^{\mathrm{\infty }}\mathrm{\mid }{u}_{nl}(r){\mathrm{\mid }}^2({\int }_{S^2}{\mathbf{1}}_B(r\mathrm{\Omega })\mathrm{\mid }{Y}_{l\bar{m}}(\mathrm{\Omega }){\mathrm{\mid }}^{2}d\sigma (\mathrm{\Omega }))dr;$ in particular, for a radial set $B=\{x:a<\mathrm{\mid }x\mathrm{\mid }<b\}$ with $0\le a<b\le +\mathrm{\infty }$ one has ${\mu }_{{\mathrm{\Psi }}_{nl\bar{m}}}^G(B)=\frac{1}{\mathrm{\parallel }{u}_{nl}{\mathrm{\parallel }}_2^2}{\int }_a^b\mathrm{\mid }{u}_{nl}(r){\mathrm{\mid }}^{2}dr,$ so the radial density $p_{nl}(r):=\mathrm{\mid }{u}_{nl}(r){\mathrm{\mid }}^2\mathrm{/}\mathrm{\parallel }{u}_{nl}{\mathrm{\parallel }}_2^2$ is a probability density on $(0,\mathrm{\infty })$: $p_{nl}\ge 0$ and ${\int }_0^{\mathrm{\infty }}{p}_{nl}(r)dr=1$;

For a general unit bound eigenvector $\chi$ with $H_{\kappa }\chi =E_n\chi$, write $\chi ={\sum }_{l=0}^{n-1}{\sum }_{\bar{m}=-l}^l{b}_{l\bar{m}}{\mathrm{\Psi }}_{nl\bar{m}}$, where ${\sum }_{l,\bar{m}}\mathrm{\mid }{b}_{l\bar{m}}{\mathrm{\mid }}^2=1$, and set ${\widehat{u}}_{nl}:=u_{nl}\mathrm{/}\mathrm{\parallel }{u}_{nl}{\mathrm{\parallel }}_2$. Then ${\mu }_{\chi }^G(B)={\int }_0^{\mathrm{\infty }}{\int }_{S^2}{\mathbf{1}}_B(r\mathrm{\Omega }){\mid {\sum }_{l=0}^{n-1}{\sum }_{\bar{m}=-l}^l{b}_{l\bar{m}}{\widehat{u}}_{nl}(r)Y_{l\bar{m}}(\mathrm{\Omega })\mid }^{2}d\sigma (\mathrm{\Omega })dr,$ and its radial marginal has density $p_{\chi }(r)={\sum }_{l=0}^{n-1}{\sum }_{\bar{m}=-l}^l\mathrm{\mid }{b}_{l\bar{m}}{\mathrm{\mid }}^2{p}_{nl}(r)\mathrm{.}$ The general position density includes interference terms and need not factor as a single $p_{nl}$ times one spherical-harmonic weight.

(b) for $n=1,l=\bar{m}=0$ the normalized vector ${\mathrm{\Psi }}_{100}$ has a representative of the form ${\mathrm{\Psi }}_{100}(x)={\zeta }_{100}(\pi a_{\kappa }^3{)}^{-1\mathrm{/}2}{e}^{-r\mathrm{/}{a}_{\kappa }},\mathrm{\mid }{\zeta }_{100}\mathrm{\mid }=1,$ and its radial density is $p_{10}(r)=4r^2{a}_{\kappa }^{-3}{e}^{-2r\mathrm{/}{a}_{\kappa }};$ the mean of $p_{10}$ is ${\int }_0^{\mathrm{\infty }}r{p}_{10}(r)dr=3a_{\kappa }\mathrm{/}2$, the unique maximum of $p_{10}$ is at $r=a_{\kappa }$, the spatial density of this representative is $\mathrm{\mid }{\mathrm{\Psi }}_{100}{\mathrm{\mid }}^2=\frac{1}{\pi a_{\kappa }^3}{e}^{-2r\mathrm{/}{a}_{\kappa }}$, strictly decreasing in $r$ and maximal at the origin, and for every $\epsilon >0$ ${\mu }_{{\mathrm{\Psi }}_{100}}^G(\{\mathrm{\mid }x\mathrm{\mid }<\epsilon \})=\frac{1}{6}(\frac{2\epsilon }{a_{\kappa }}{)}^3+O(\left(\frac{\epsilon }{a_{\kappa }}{)}^4\right)>0,$ which tends to $0$ as $\epsilon \downarrow 0$: the density maximum at the origin is not an atom of probability at any point;

(c) for every label the reduced solution has the polynomial-exponential form $u_{nl}(r)=c_{nl}{r}^{l+1}{e}^{-r\mathrm{/}(n{a}_{\kappa })}{L}_{n-l-1}^{(2l+1)}(2r\mathrm{/}(n{a}_{\kappa }))(c_{nl}\mathrm{\ne }0),$ with $L_j^{(a)}$ the Laguerre polynomial of integer order, so the radial density of the level $(n,l)$ is a polynomial times $r^{2l+2}$ times a decreasing exponential, and its normalisation constant is fixed by ${\int }_0^{\mathrm{\infty }}\mathrm{\mid }{u}_{nl}{\mathrm{\mid }}^{2}dr=\mathrm{\parallel }{\psi }_{nl\bar{m}}{\mathrm{\parallel }}^2$;

(d) Reading and scope. The statements are conditional on P1, P2, P4, P5, the declared Coulomb model and the cited mathematics. The radial density is the reading of the model's position observable on radial sets; it is not a trajectory, an orbit or a definite distance, and the model assigns positive probability to every open ball around the origin and to arbitrarily large radii. No measured radius, no measured density and no identification of $a_{\kappa }$ with a laboratory length is asserted; numerical comparison requires the measured constants as qualified empirical inputs and is deferred.

• The radial density is not a radial observable. The quantity $p_{nl}$ is the disintegration of the position law on radial Borel sets, a reading of the declared position observable; the model does not assign the radius as an independent sharp observable, and no commutation or measurement claim about a radius operator is made.
• Scales. The two length scales $a_{\kappa }$ and the energy scale ${\mathrm{\hslash }}^2\mathrm{/}(2m{a}_{\kappa }^2)$ are model data fixed by $\mathrm{\hslash },m,\kappa$; the identification of $a_{\kappa }$ with a physical Bohr radius of a hydrogen atom is the tested identification of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clause (e) and carries its uncertainty.