Qm Central Potentials And Hydrogen — Examples

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.

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 $a:=a_{\kappa }={\mathrm{\hslash }}^2\mathrm{/}(m\kappa )$, and the radial reading of The one-centre Coulomb model: position and radial distributions of the bound states, with the ground-state density and its scales. In the notation of that item, $u_{nl}$ denotes the reduced regular solution with ${\psi }_{nlm}(x)=u_{nl}(r)Y_{lm}(\mathrm{\Omega })\mathrm{/}r$ and $p_{nl}=\mathrm{\mid }{u}_{nl}{\mathrm{\mid }}^2\mathrm{/}\mathrm{\parallel }{u}_{nl}{\mathrm{\parallel }}_2^2$ the radial density of the normalised bound state. For the three reduced solutions below choose the normalisation $\mathrm{\parallel }{u}_{nl}{\mathrm{\parallel }}_2=1$ and the phase for which the coefficient $c_{nl}$ multiplying the displayed Laguerre polynomial-exponential expression is positive real. Then:

(a) $u_{10}(r)=2a^{-3\mathrm{/}2}r{e}^{-r\mathrm{/}a}$ and $u_{20}(r)=2^{-3\mathrm{/}2}{a}^{-3\mathrm{/}2}(2r-\frac{r^2}{a})e^{-r\mathrm{/}(2a)}$ and $u_{21}(r)=2^{-3\mathrm{/}2}{3}^{-1\mathrm{/}2}{a}^{-5\mathrm{/}2}{r}^2{e}^{-r\mathrm{/}(2a)}$;

(b) $\mathrm{\parallel }{u}_{10}{\mathrm{\parallel }}_2=\mathrm{\parallel }{u}_{20}{\mathrm{\parallel }}_2=\mathrm{\parallel }{u}_{21}{\mathrm{\parallel }}_2=1$, so for these states the reduced solution is already normalised and $p_{nl}=\mathrm{\mid }{u}_{nl}{\mathrm{\mid }}^2$;

(c) the moments ${\int }_0^{\mathrm{\infty }}r{p}_{10}(r)dr=\frac{3a}{2},{\int }_0^{\mathrm{\infty }}{r}^2{p}_{10}(r)dr=3a^2,{\int }_0^{\mathrm{\infty }}r{p}_{20}(r)dr=6a,{\int }_0^{\mathrm{\infty }}r{p}_{21}(r)dr=5a,$ and ${\int }_0^{\mathrm{\infty }}\frac{p_{20}(r)}{r}dr={\int }_0^{\mathrm{\infty }}\frac{p_{21}(r)}{r}dr=\frac{1}{4a};$

(d) the same-$l$ pair is orthogonal, ${\int }_0^{\mathrm{\infty }}{u}_{10}(r)u_{20}(r)dr=0$, while the different-$l$ pair is not: ${\int }_0^{\mathrm{\infty }}{u}_{20}(r)u_{21}(r)dr=-\frac{\sqrt{3}}{2}\mathrm{\ne }0$, so the reduced radial functions of different angular momentum are not orthogonal in $L^2((0,\mathrm{\infty }),dr)$, the orthogonality of the full bound states being carried by the angular factor;

(e) the closed forms of (a) agree with the standard radial functions $R_{nl}=u_{nl}\mathrm{/}r$ of the cited treatments for these states, and the value $\int r{p}_{nl}$ for the circular state $n=2$, $l=1$ agrees with the general circular-orbit formula $⟨r{⟩}_{n,n-1}=n(n+\frac{1}{2})a$ at $n=2$;

(f) Scope. The computation is conditional on the declared model and the cited mathematics; $a$ is a model scale, the moments are moments of the model's radial reading and not measured radii, and no measurement, orbit or trajectory is asserted.

• Why cross-$l$ overlaps need not vanish. The full bound states ${\psi }_{nlm}$ and ${\psi }_{n^{\mathrm{\prime }}{l}^{\mathrm{\prime }}{m}^{\mathrm{\prime }}}$ are orthogonal in $L^2({\mathbb{R}}^3)$ whenever the angular factors $Y_{lm},Y_{l^{\mathrm{\prime }}{m}^{\mathrm{\prime }}}$ are orthogonal, even if the reduced radial functions overlap; the nonzero value $-\sqrt{3}\mathrm{/}2$ computed in (d) is exactly such a case. No orthogonality of reduced functions across angular momenta is claimed anywhere on this pair.
• Moments are readings, not measurements. The mean radius and the inverse radius are moments of the model's radial reading; the inverse-radius value $1\mathrm{/}(4a)$ for the two $n=2$ states is a consistency check with the model's virial-type relation and is not compared with any measured quantity here.

• What the pair's spectral statements are. The model's content is the conditional spectral statement of The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum: bound energies $E_n=-\frac{m{\kappa }^2}{2{\mathrm{\hslash }}^2{n}^2}$ with $n^2$-fold model degeneracy, a complete negative spectral subspace and a nonzero continuum, together with the two-body threshold reading of The declared two-body Coulomb model: reduced-mass decomposition, relative levels as thresholds and the absence of stationary total states. These are statements about the declared model and its assigned parameters; they are not statements about any measured hydrogen line.

• Why no experiment appears on this pair. The design reservation for this pair names a spectroscopy data candidate and marks it pending. No source-backed observation record with preparation, apparatus, calibration, sampling and uncertainty was harvested for this batch, so no experiment item is built and no measured wavelength, frequency or transition is reported. The source treatments do describe the Lyman, Balmer, Paschen and Brackett series and the historical Balmer fit; this page neither reproduces their numerical values nor treats them as its own observations, and the comparison of the model with measured spectra is deferred to a future experiment item carrying its own uncertainty record.

• What a comparison would presuppose. An emission-frequency illustration, conditional on an initial model energy $E_i$ greater than the final energy $E_f$, uses $\nu =(E_i-E_f)\mathrm{/}h$ together with the radiation extension and the light-matter coupling of later pairs; no transition-frequency claim is made on this page. Any comparison with a measured line would test the combined model, apparatus and constant assumptions: the preparation and isolation idealization, the identification of the species parameters with measured masses and charges, the treatment of linewidth, Doppler and instrumental response, and the sampling model of the recorded sample. Agreement of a finite set of measured lines with the predicted values would be evidence about the combined assumptions; it would not be a deductive proof of the model, and disagreement would require the same auxiliary assumptions to be examined before the model is rejected.

• Excluded corrections and their status. The gross-structure model omits spin, relativistic fine structure, finite nuclear size, QED corrections including the Lamb shift, hyperfine structure and shielding, as recorded in The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clause (f); the source treatments describe some of these corrections and their consequences, but no magnitude, splitting or numerical value is asserted or used by this pair.

• Identification caution. The assignment of the model's parameters to a laboratory hydrogen atom is the tested identification of clause (e) of the model definition under the preparation and apparatus postulate (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions); it is not derived from the mathematics of the model and carries the uncertainty of that identification. The model's spectral statements remain conditional until such an identification, with its qualifications, is supplied.

• What the page contains. Two conditional computations and one status record in the declared Coulomb model: the exact equivalence of the bound-energy parametrisations in terms of $(m,\kappa )$, $(\mathrm{\hslash },m,a_{\kappa })$ and $(m,\alpha ,c)$, the reduced-mass factor between the relative and fixed-nucleus energies, and the dimensional checks of those identities (Bound-energy parametrisations of the Coulomb model: exact identities, units and the reduced-mass factor); the explicit reduced radial functions, normalisations and low moments for the states with $n\le 2$, including the check that reduced radial functions of different angular momentum are not orthogonal (Low-lying radial distributions of the Coulomb model: normalisations, moments and the failure of cross-l orthogonality of reduced functions); and the empirical-status record deferring the comparison with measured spectra (Empirical status of the hydrogen spectrum on this pair: pending spectroscopy evidence and the limits of the model's spectral statements).

• What the page does not contain. No experiment, no reported line, no wavelength or frequency value, no count, no calibration and no statistical inference appears; no item assigns a numerical value to any model parameter or measured constant, and none of the computations establishes that the model or its postulates describes nature. The identification of a concrete hydrogen atom with the declared model is a tested identification under the preparation and apparatus postulate (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions) and is not made here.

• Dependence on the A page. Every statement of the page is conditional on the declared Coulomb model, its assigned energy observable and the spectral theorem (The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions, The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum); the model scales $a_{\kappa }={\mathrm{\hslash }}^2\mathrm{/}(m\kappa )$ and the energies are model data, not measured lengths or energies.

• Deferred content. Transition frequencies, line intensities, selection rules and photon statements belong to the radiation and light-matter items of the quantum-mechanics track; scattering and ionisation belong to the three-dimensional scattering pair. The numerical spectroscopic comparison remains a pending experiment candidate and is not anticipated by any claim of this page.