Bound-energy parametrisations of the Coulomb model: exact identities, units and the reduced-mass factor

For this example take the equal-magnitude, opposite-charge branch $q_{-}=q_{+}=q$ of clause (d) of the declared Coulomb model. When the example is read as hydrogen, the hydrogen assignment is $Z=1$, so $q=e$ and $\kappa =e^2\mathrm{/}(4\pi {\epsilon }_0)$; for a general hydrogen-like ion the charge product is $q_{-}{q}_{+}=Z{e}^2$ and must replace $q^2$ in $\kappa$. Adopt 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 $\mathrm{\hslash }>0$, $m>0$, coupling $\kappa >0$ and length scale $a_{\kappa }={\mathrm{\hslash }}^2\mathrm{/}(m\kappa )$, the bound-energy result 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 two-body reading The declared two-body Coulomb model: reduced-mass decomposition, relative levels as thresholds and the absence of stationary total states with masses $m_1,m_2>0$ and reduced mass $\mu =m_1{m}_2\mathrm{/}(m_1+m_2)$. Let the charge parameter be $q>0$ with $\kappa =q^2\mathrm{/}(4\pi {\epsilon }_0)$, let $c$ be the speed of light of the SI conventions of Dimensions, SI unit conventions and quantity types of the framework clause (c), and put $\alpha :=\frac{q^2}{4\pi {\epsilon }_0\mathrm{\hslash }c}=\frac{\kappa }{\mathrm{\hslash }c},$ the dimensionless coupling of the model. Then the following identities hold exactly for the declared parameters:

(a) $E_n:=-{\displaystyle \frac{m{\kappa }^2}{2{\mathrm{\hslash }}^2{n}^2}}=-{\displaystyle \frac{{\mathrm{\hslash }}^2}{2m{a}_{\kappa }^2{n}^2}}$ for every $n\ge 1$, so ${\displaystyle \frac{m{\kappa }^2}{2{\mathrm{\hslash }}^2}}={\displaystyle \frac{{\mathrm{\hslash }}^2}{2m{a}_{\kappa }^2}}$ and $E_1=-{\displaystyle \frac{{\mathrm{\hslash }}^2}{2m{a}_{\kappa }^2}}$;

(b) $\kappa =\alpha \mathrm{\hslash }c$, hence $E_n=-{\displaystyle \frac{m{\alpha }^2{c}^2}{2n^2}}$ and $a_{\kappa }={\displaystyle \frac{\mathrm{\hslash }}{m\alpha c}}$;

(c) the relative bound energies of the two-body model are ${\mathcal{E}}_n=-{\displaystyle \frac{\mu {\kappa }^2}{2{\mathrm{\hslash }}^2{n}^2}}={\displaystyle \frac{\mu }{m_1}}(-{\displaystyle \frac{m_1{\kappa }^2}{2{\mathrm{\hslash }}^2{n}^2}})$ with ${\displaystyle \frac{\mu }{m_1}}={\displaystyle \frac{1}{1+m_1\mathrm{/}{m}_2}}\in (0,1)$;

(d) in the reference units of Dimensions, SI unit conventions and quantity types of the framework clause (b), $[\mathrm{\hslash }]=\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, $[\kappa ]=\mathrm{k}\mathrm{g}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, $[a_{\kappa }]=\mathrm{m}$, $[E_n]=\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-2}=\mathrm{J}$ and $[\alpha ]=1$, so every displayed identity is dimensionally consistent;

(e) the bottom of the continuous spectrum of the total two-body operator is ${\mathcal{E}}_1=-{\displaystyle \frac{\mu {\kappa }^2}{2{\mathrm{\hslash }}^2}}$, which coincides with the relative ground energy, and the total operator has no eigenvalue there.

The identities are conditional model computations in terms of declared parameters; no numerical value of any parameter is assigned, and no measured quantity is reported. A numerical evaluation of $E_1$ requires measured values of constants such as $\alpha$ and of the mass ratio $m_1\mathrm{/}{m}_2$ as qualified empirical premises; it is not performed here.

• Why no numbers appear. Evaluating $E_1$ in electronvolts would require measured inputs such as the fine-structure constant and the electron-to-proton mass ratio; the model's own parameters are not measured by this page, and the empirical premises, their uncertainties and the comparison with observed spectra belong to a dedicated experiment item, which does not yet exist.
• Fixed-nucleus reference. The bracket in (c) is the fixed-nucleus energy of the one-centre model with mass parameter $m_1$; it is a reference quantity of the declared limiting reading and not a measured quantity, and the limit $m_2\to \mathrm{\infty }$ is a declared reading, not an operator convergence theorem.