The Hydrogen Atom
Hydrogen is the atom the Schrödinger equation solves exactly, and the one against which almost everything else in quantum physics is calibrated. One electron in the Coulomb field of one proton gives the spectrum \(E_{n}=-13.6\,\mathrm{eV}/n^{2}\), a degeneracy of \(n^{2}\) that is larger than rotational invariance alone can explain, and a length scale—the Bohr radius \(a_{0}=5.29\times 10^{-11}\,\mathrm{m}\)—built from \(\hbar\), the electron mass and the elementary charge. This chapter solves the problem twice: once in wave mechanics, following Schrödinger's first communication [Schroedinger:1926a], and once algebraically through the conserved Runge–Lenz vector, following Pauli [Pauli:1926], whose calculation preceded the wave equation. The second route explains the extra degeneracy: the Coulomb problem has a hidden \(\SO(4)\) symmetry, identified by Fock [Fock:1935] and Bargmann [Bargmann:1936], and the accidental degeneracy is not accidental at all.
The chapter then peels the exact solution apart layer by layer, because each correction to it is a separate piece of physics measured to its own precision: relativistic and spin–orbit fine structure, closed by the Dirac equation of The Dirac Equation; the Lamb shift [Lamb:1947], which splits levels the Dirac theory holds degenerate and thereby started renormalized quantum electrodynamics (Quantum Electrodynamics and Renormalization); the hyperfine coupling to the proton spin, whose ground-state splitting radiates the \(21\,\mathrm{cm}\) line [Ewen:1951] [Muller:1951] that mapped the Galaxy; and the finite size of the proton, read off from muonic hydrogen [Pohl:2010] [Antognini:2013]. Behaviour in external fields—the Zeeman and Stark effects—is worked as the standard application of perturbation theory in Approximation Methods and only summarized here. Because the theory is exact and the experiments are the most precise in atomic physics, hydrogen is where the fundamental constants are fixed [Tiesinga:2021] [Mohr:2025] and where a discrepancy of a few parts in \(10^{5}\) counts as a crisis.
The Hydrogen Atom: all derivations of this chapter are pending.
The spectrum before the theory
Balmer's formula and the Rydberg constant
[Reserved: the four visible hydrogen lines and Balmer's empirical fit \(\lambda=hm^{2}/(m^{2}-4)\) [Balmer:1885]; Rydberg's generalization to a difference of two terms, valid across the alkali spectra as well [Rydberg:1890], and the Rydberg–Ritz combination principle [Ritz:1908], which made term differences rather than lines the primitive objects; the named series and the modern value \(R_{\infty}=1.0973731568\times 10^{7}\,/\mathrm{m}\) [Tiesinga:2021]. What these formulas were: an exact, purely empirical regularity with no mechanism whatever, and the standing challenge that Atomic Models and Spectra recounts.]
The visible emission spectrum of atomic hydrogen is neither a continuum nor an arbitrary set of lines. Balmer found that the four lines then measured obey
with the single constant \(b=3.6456\times 10^{-7}\,\mathrm{m}\), and used Equation (81.1) to predict further members of the series that were afterwards found [Balmer:1885]. Rydberg showed that the same regularity governs the whole hydrogen spectrum, and the alkali spectra besides, in the form
with one constant serving every series of the atom [Rydberg:1890]; Equation (81.1) is the case \(n_{1}=2\). The constant is now \(R_{\infty}=1.0973731568\times 10^{7}\,/\mathrm{m}\), one of the most accurately known quantities in physics [Mohr:2025]. Classical physics predicts no discrete spectrum at all, still less one indexed by two integers, and least of all a radiating electron that does not spiral into the nucleus.
The bound-state eigenvalues of the Coulomb problem: the radial equation for the potential that falls as the inverse first power of the distance, the termination of its series solution which quantizes the energy as minus a constant over the square of an integer, and the identification of that constant with a combination of the electron mass, the elementary charge and Planck's constant
The wavenumbers of the lines of an atom are not independent of one another. Each is a difference of two members of a single list of terms characteristic of the atom, so that the sum of the wavenumbers of two observed lines is very often itself the wavenumber of an observed line [Ritz:1908]. The rule holds for atoms whose spectra no formula of Balmer's kind describes, it was established empirically two decades before any mechanism for it existed, and it is what made the terms, rather than the lines, the primitive objects of spectroscopy.
Derivation. Assume, with the postulates of The Postulates of Quantum Mechanics, that the atom possesses a set of stationary states of energies \(E_{i}\), and that radiation is emitted or absorbed only in transitions between them, one photon of energy \(h\nu=E_{i}-E_{f}\) per transition. Define the term of a state as \(T_{i}:=-E_{i}/hc\), a positive number for a bound state. The wavenumber of the line joining states \(i\) and \(f\) is then
a difference of two members of one list, as observed; and for any three states,
so that two lines sharing a level combine into a third. That third line is observed only when the transition from \(i\) to \(g\) is itself allowed by the selection rules of Angular Momentum and Spin, which is exactly why the empirical rule holds very often rather than always — a discrepancy that was noticed long before it was explained. Equation (81.2) is the special case of Equation (81.3) in which the terms of one atom happen to form the sequence \(T_{n}=R_{\infty}/n^{2}\); the content of Phenomenon 81.2 is that terms exist at all, which is a statement about stationary states and holds whatever the sequence.
∎What the old quantum theory got right
[Reserved: Bohr's quantized orbits [Bohr:1913] reproducing Balmer's formula and predicting \(R_{\infty}\) from \(\hbar\), \(m\) and \(e\); Sommerfeld's elliptical orbits and the relativistic fine-structure formula in \(\alpha^{2}\) [Sommerfeld:1916], numerically right for reasons later shown to be wrong; the Bohr–Sommerfeld quantization as the action condition of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy; and the failures that ended the programme—no helium, no intensities, no ground for the selection rules. Detailed account in Atomic Models and Spectra.]
The exact solution
Two bodies, one Coulomb potential
[Reserved: separation of the centre of mass, the reduced mass \(\mu=m_{\mathrm{e}}m_{\mathrm{p}}/(m_{\mathrm{e}}+m_{\mathrm{p}})\), and the resulting mass-dependent shift of every level; the isotope shift as its measurable consequence, and the discovery of deuterium by exactly that shift in the Balmer lines [Urey:1932]; the Coulomb potential taken from Electrostatics and its \(1/r\) form as the input that everything below depends on; why the same reduction serves positronium and muonic hydrogen in Section 81.7.2.]
Every line of the hydrogen spectrum has a faint companion displaced to shorter wavelength by a fixed relative amount of about \(2.7\times10^{-4}\); for the red Balmer line at \(656.3\,\mathrm{nm}\) the companion stands about \(0.18\,\mathrm{nm}\) away. Urey, Brickwedde and Murphy found the companions, showed that they strengthened when the sample was enriched by evaporating liquid hydrogen nearly to exhaustion, and identified them as the spectrum of an isotope of mass number two [Urey:1932]. The nucleus is therefore not an infinitely heavy centre of force: the spectrum knows the nuclear mass, and the amount by which it knows it is the electron-to-nucleus mass ratio.
Derivation. Separating the centre of mass of two bodies interacting through a potential that depends only on their separation leaves a one-body problem in the relative coordinate, with the reduced mass \(\mu=m_{\mathrm{e}}M/\left(m_{\mathrm{e}}+M\right)\) in place of the electron mass, \(M\) being the nuclear mass. Every energy of the Coulomb problem is proportional to the mass appearing in the kinetic term, so the Rydberg constant of a nucleus of mass \(M\) is
and wavelengths, being inversely proportional to it, satisfy \(\lambda_{M}=\lambda_{\infty}\left(1+m_{\mathrm{e}}/M\right)\) to first order in the small ratio. Comparing a nucleus of one proton with one of twice that mass,
with the mass ratio taken from [Mohr:2025]. Applied to the red Balmer line this is a displacement of \(0.18\,\mathrm{nm}\), the observed one; and it falls on the short side because the heavier nucleus gives the larger reduced mass, hence by Equation (81.4) the larger Rydberg constant and the larger wavenumber. The same substitution run to its limit gives positronium, whose reduced mass is \(m_{\mathrm{e}}/2\) and whose levels are therefore half of hydrogen's (Section 81.7.2).
∎The radial equation and its quantization
[Reserved: separation in spherical coordinates using the angular solution of Angular Momentum and Spin; the effective radial equation with the centrifugal barrier; the asymptotic factorization \(\ee^{-\rho/2}\rho^{\ell+1}\) and the termination of the series that quantizes the energy; associated Laguerre polynomials, the principal quantum number \(n\ge\ell+1\), and the spectrum \(E_{n}=-\mu e^{4}/(32\pi^{2}\varepsilon_{0}^{2}\hbar^{2}n^{2})\) [Schroedinger:1926a]. The radial problem as a Sturm–Liouville system, with node counting, is Ordinary Differential Equations and Sturm–Liouville Theory.]
Eigenfunctions, orbitals and expectation values
[Reserved: the normalized wavefunctions \(\psi_{n\ell m}=R_{n\ell}(r)Y_{\ell m}(\theta,\varphi)\); radial probability densities and their nodes; \(\avg{r}\), \(\avg{1/r}\) and \(\avg{1/r^{3}}\), the last of which is what the fine-structure and hyperfine calculations below actually need; the \(\ell=0\) states alone non-vanishing at the origin, which is why they alone carry the Fermi contact and Lamb terms; virial theorem \(\avg{T}=-\tfrac{1}{2}\avg{V}\); the continuum states above threshold and the photoionization edge (Radiation and Scattering of Electromagnetic Waves).]
Levels, degeneracy and scales
[Reserved: the level diagram, the ionization energy \(13.6\,\mathrm{eV}\), the Bohr radius as \(\avg{r}\) of the ground state, and the fine-structure constant \(\alpha\) entering as the ratio of the electron's orbital velocity to \(c\)—which is why every correction below arrives as a further power of \(\alpha\), and why the numerical value of \(\alpha\) is fixed by measurements of exactly this kind (Experiment: Precision Spectroscopy and Atomic Clocks). The degeneracy: \(2\ell+1\) from rotational invariance, but a total of \(n^{2}\) orbital states independent of \(\ell\), which no rotational argument gives. That surplus is the subject of the next section.]
To the accuracy at which fine structure may be neglected, the bound levels of hydrogen depend only on the principal quantum number and not on the orbital one: all the terms with \(\ell=0,1,\ldots,n-1\) belonging to one \(n\) coincide, giving \(\sum_{\ell=0}^{n-1}\left(2\ell+1\right)=n^{2}\) degenerate orbital states. Rotational invariance guarantees only the \(2\ell+1\) states of a single \(\ell\), so the surplus is unexplained by any symmetry so far introduced. Two observations show that the coincidence is exact and not merely close. Hydrogen alone among atoms has a Stark effect linear in the applied electric field [Stark:1914], and a linear response requires states of opposite parity to be exactly degenerate, since a field mixes them in first order only when it costs no energy to do so. And the alkali spectra, whose inner electrons screen the nucleus and spoil the exact inverse-square-law force, show precisely the term splittings by \(\ell\) that hydrogen lacks [Rydberg:1890]: it is the \(1/r\) potential that is responsible, not the one-electron character of the atom.
The \(\ell\)-independence of the Coulomb spectrum: the conserved Runge–Lenz vector peculiar to the inverse-square-law force, the closure of its commutators with the angular momentum on the algebra of \(\SO(4)\) for bound states, and the identification of the levels with irreducible representations of that group whose dimension is the square of the principal quantum number
Degeneracy and the hidden symmetry
The Runge–Lenz vector and the algebraic solution
[Reserved: the classical conserved vector of the Kepler problem, pointing along the major axis and expressing the closure of the orbits (Central Forces and Statics), reintroduced into the quantum theory by Lenz [Lenz:1924]; its Hermitian symmetrization \(\vect{A}=\tfrac{1}{2\mu}(\vect{p}\times\vect{L}-\vect{L}\times\vect{p}) -e^{2}\vect{r}/(4\pi\varepsilon_{0}r)\), the commutators of \(\vect{A}\) with \(\vect{L}\) and with itself, and Pauli's derivation of the whole spectrum from that algebra alone [Pauli:1926]—published before the wave equation and reproducing Balmer exactly.]
$\SO(4)$ and Fock's construction
[Reserved: the rescaled generators closing on \(\mathfrak{so}(4)\cong\mathfrak{su}(2)\oplus\mathfrak{su}(2)\) for bound states (and \(\mathfrak{so}(3,1)\) for scattering states); Fock's demonstration that stereographic projection of momentum space onto the three-sphere turns the bound-state equation into the equation for spherical harmonics on \(S^{3}\), so that the \(n^{2}\) degeneracy is the dimension of an \(\SO(4)\) irreducible representation [Fock:1935], with the algebraic completion by Bargmann [Bargmann:1936]. Group-theoretic setting: Linear Algebra and Representation Theory and Lie Groups, Lie Algebras, and Fibre Bundles. The lesson—that an unexplained degeneracy signals a symmetry not yet identified—is used again for the isotropic oscillator and in Quantum Chromodynamics.]
Fine and hyperfine structure
Relativistic corrections and spin–orbit coupling
[Reserved: the three terms of order \(\alpha^{2}\) relative to the Bohr energies—the relativistic kinetic correction, the spin–orbit coupling with its Thomas factor (Angular Momentum and Spin) [Thomas:1926], and the Darwin contact term for \(\ell=0\) [Darwin:1928]; their sum depending only on \(n\) and \(j\), so that \(2s_{1/2}\) and \(2p_{1/2}\) remain degenerate; the exact Dirac result reproducing Sommerfeld's formula [Sommerfeld:1916] [Dirac:1928] and derived in The Dirac Equation; the observed \(2p_{3/2}\)–\(2p_{1/2}\) splitting of about \(10.9\,\mathrm{GHz}\) as the measured scale.]
The Lamb shift
[Reserved: the microwave measurement showing \(2s_{1/2}\) lying about \(1058\,\mathrm{MHz}\) above \(2p_{1/2}\), in flat contradiction with the Dirac degeneracy [Lamb:1947]; Bethe's non-relativistic estimate, the first successful renormalized calculation, obtained within weeks [Bethe:1947]; the physical content—vacuum polarization and the electron's self-interaction with its own radiation field—and the full treatment in Quantum Electrodynamics and Renormalization; the modern value and its role, together with the electron \(g-2\) (Experiment: The Electron Anomalous Magnetic Moment), as the sharpest confrontation between quantum field theory and measurement.]
The states \(2s_{1/2}\) and \(2p_{1/2}\) of hydrogen are not degenerate, although the Dirac theory makes them exactly so, its energies depending only on \(n\) and \(j\). Driving microwave transitions directly between the two in a beam of metastable atoms, and using the metastable atoms themselves as the detector, Lamb and Retherford found \(2s_{1/2}\) lying above \(2p_{1/2}\) by about \(1000\,\mathrm{MHz}\) [Lamb:1947]; the modern value is close to \(1058\,\mathrm{MHz}\). The shift is only about one part in \(10^{6}\) of the binding energy of the level, and it falsified the best relativistic one-particle theory then available. Bethe's estimate of it, made within weeks, was the first calculation in which the infinities of the radiation theory were removed by subtracting a free-particle self-energy [Bethe:1947].
The radiative level shift: the interaction of the bound electron with the fluctuating quantized electromagnetic field, the difference between the bound and free self-energies which is what remains finite after regularization, and the vacuum-polarization contribution of opposite sign that accompanies it
Hyperfine structure
[Reserved: the coupling of the electron spin to the proton magnetic moment, dominated for \(\ell=0\) by the Fermi contact term proportional to \(\abs{\psi(0)}^{2}\) [Fermi:1930]; the total \(\vect{F}=\vect{I}+\vect{J}\) and the singlet–triplet splitting of the ground state, smaller than the fine structure by the electron-to-proton mass ratio—about \(5.9\,\mu\mathrm{eV}\), that is \(1420.4058\,\mathrm{MHz}\); the proton magnetic moment measured independently by magnetic resonance (Angular Momentum and Spin); the hyperfine anomaly as a probe of nuclear magnetization distribution (Nuclear Forces and Nuclear Structure).]
Hydrogen in external fields
[Reserved: a summary, with the derivations deliberately placed in Approximation Methods where the perturbative machinery is built—the linear Stark effect, peculiar to hydrogen because the \(\ell\) degeneracy of Section 81.2.4 survives [Stark:1914] [Schroedinger:1926c]; the Zeeman effect in the weak field with Landé's \(g\) factor and the Paschen–Back regime in the strong field [Zeeman:1897] [Paschen:1912]; diamagnetic hydrogen in very strong fields as a laboratory for quantum chaos (Nonlinear Dynamics and Chaos).]
The 21-centimetre line
A prediction made before the instrument existed
[Reserved: van de Hulst's wartime calculation that the hyperfine ground-state transition of neutral hydrogen should be detectable as a radio line near \(21\,\mathrm{cm}\), despite a spontaneous-emission lifetime of order \(10^{7}\) years, because the Galaxy contains enough atoms [vandeHulst:1945]; the magnetic-dipole character of the transition and the selection rules of Angular Momentum and Spin that make it so slow; why a line this narrow and this weak is nonetheless the most useful in radio astronomy.]
Detection
[Reserved: the independent detections announced together in 1951—Ewen and Purcell at Harvard [Ewen:1951] and Muller and Oort at Kootwijk [Muller:1951]—with the receiver technique, the frequency-switching that pulled the line out of the noise, and the measured rest frequency now known to \(1420.4058\,\mathrm{MHz}\). This is the worked case of a quantum-mechanical energy difference of a few \(\mu\mathrm{eV}\) becoming an astronomical instrument.]
Neutral atomic hydrogen in its electronic ground state radiates one spectral line in the radio band, at a rest frequency of \(1420.4058\,\mathrm{MHz}\) — a vacuum wavelength close to \(21\,\mathrm{cm}\). Ewen and Purcell detected it in emission from the Milky Way with a horn antenna mounted out of a laboratory window and a receiver that switched frequency rapidly so as to subtract its own noise [Ewen:1951]; Muller and Oort confirmed it within weeks at Kootwijk and began mapping the Galaxy with it [Muller:1951]. The transition joins the two hyperfine components of the ground state, split by the interaction of the electron's magnetic moment with the proton's. The splitting is about \(5.9\,\mu\mathrm{eV}\), smaller than the fine structure by roughly the electron-to-proton mass ratio, and the transition is a magnetic dipole one with a spontaneous lifetime of order \(10^{7}\) years — so weak that nothing smaller than a galaxy could have been the source in which it was first seen.
The ground-state hyperfine splitting of hydrogen: the Fermi contact interaction between the electron and nuclear magnetic moments, its expectation value in the ground state, which is proportional to the probability density of the electron at the origin, and the resulting separation of the singlet from the triplet
What the line measures
[Reserved: Doppler-shifted \(21\,\mathrm{cm}\) emission mapping the rotation curve and spiral structure of the Galaxy [Muller:1951]; column densities of neutral hydrogen; the flat rotation curves of spiral galaxies as one of the primary lines of evidence catalogued in The Dark Sector: Evidence Without Explanation; absorption against background sources; and the redshifted line as a probe of the neutral intergalactic medium (Evidence-Based Cosmology). The complementary Lyman-\(\alpha\) absorption forest in quasar spectra [Lynds:1971] uses the \(n=1\to2\) transition of the same atom.]
Precision hydrogen and the constants
The Rydberg constant and optical frequency metrology
[Reserved: two-photon Doppler-free spectroscopy of the \(1s\)–\(2s\) transition, whose \(1\,\mathrm{Hz}\)-scale natural width makes it one of the sharpest resonances in physics; the measurement \(2.4661\times 10^{15}\,\mathrm{Hz}\) to a few parts in \(10^{15}\) using an optical frequency comb referenced to a caesium standard [Parthey:2011]; the least-squares adjustment in which hydrogen data, together with the proton radius, determine \(R_{\infty}\) [Tiesinga:2021] [Mohr:2025]; the SI traceability chain of Measurement, SI Units, and the Theory of Errors and the experiment chapter Experiment: Precision Spectroscopy and Atomic Clocks.]
The proton radius
[Reserved: the finite-size correction shifting \(s\) levels in proportion to the mean square charge radius; muonic hydrogen, in which the muon's larger mass concentrates the wavefunction and amplifies the effect by roughly \((m_{\mu}/m_{\mathrm{e}})^{3}\), giving \(r_{\mathrm{p}}=0.84184(67)\,\mathrm{fm}\) [Pohl:2010], confirmed and sharpened by the \(2s\)–\(2p\) transitions of [Antognini:2013]—about \(7\sigma\) below the then-accepted electronic and scattering value: the “proton radius puzzle”. Its resolution by new electronic-hydrogen measurements [Beyer:2017] [Bezginov:2019] and the revised CODATA value [Mohr:2025]; the honest reading, that a \(7\sigma\) discrepancy was settled by remeasurement rather than by new physics.]
The \(s\) levels of hydrogen lie slightly higher than a point nucleus would place them, by an amount proportional to the mean square charge radius of the proton, and the excess is large enough to be measured. Replacing the electron by a muon concentrates the wavefunction on the nucleus and amplifies the effect by millions: laser spectroscopy of the \(2s\)–\(2p\) transitions of muonic hydrogen gave \(r_{\mathrm{p}}=0.84184(67)\,\mathrm{fm}\) [Pohl:2010], and a second transition sharpened it [Antognini:2013]. That value sat about \(7\sigma\) below the electronic and electron-scattering value then accepted, and the discrepancy was closed by remeasuring the electronic transitions rather than by new physics [Beyer:2017] [Bezginov:2019] [Mohr:2025]. A property of the nucleus is thus read out of the energy levels of the electron cloud around it.
Derivation. Only the amplification factor is derived here; the coefficient belongs with the perturbation theory of Approximation Methods. Inside a nucleus of finite size the potential departs from the point-charge Coulomb form by some \(\delta V(r)\), positive because the enclosed charge is reduced, and supported on a region of the size of the proton. To first order the shift of a level is \(\Delta E=\int\abs{\psi}^{2}\,\delta V\,\dd^{3}r\), and since that region is smaller than the atom by five orders of magnitude, \(\abs{\psi}^{2}\) may be replaced by its value at the origin:
Two consequences follow. Only \(\ell=0\) states shift at all, because \(\psi(0)=0\) for every \(\ell>0\) — which is why the muonic measurement compares an \(s\) level with a \(p\) level. And \(\abs{\psi(0)}^{2}=1/\pi a^{3}\) for the ground state, with the Bohr radius \(a=4\pi\varepsilon_{0}\hbar^{2}/\mu e^{2}\) inversely proportional to the reduced mass, so that \(\Delta E\propto\mu^{3}\avg{r_{\mathrm{p}}^{2}}\). Muonic hydrogen has \(\mu\simeq186\,m_{\mathrm{e}}\), so its finite-size shift exceeds the electronic one by a factor of about \(6\times10^{6}\). This is why one laser transition in an exotic atom could outweigh decades of electronic spectroscopy, and equally why a systematic error in that one measurement would have had nothing to check it against — which is the reason the discrepancy was taken seriously rather than dismissed.
∎Antihydrogen and the tests it carries
[Reserved: trapped antihydrogen and laser spectroscopy of its \(1s\)–\(2s\) transition, agreeing with hydrogen to a relative precision of about \(2\times10^{-12}\) [Ahmadi:2018]—at present the sharpest spectroscopic test of CPT invariance (Discrete Symmetries and CPT); the antihydrogen hyperfine structure; and the gravitational free-fall measurement, which belongs with the equivalence-principle tests of The Equivalence Principle and Classical Tests. Positron and antiproton production and the trapping technique are the experimental content.]
Hydrogen-like systems
One electron, other cores
[Reserved: hydrogenic ions \(Z^{2}\)-scaled, up to hydrogen-like uranium where \(Z\alpha\) approaches unity and the expansion in Section 81.4.1 fails; Rydberg atoms, in which a single highly excited electron sees a screened core and reproduces the hydrogen spectrum with a quantum defect, with polarizabilities scaling as \(n^{7}\) [Gallagher:1994] (used in the quantum optics of Quantum Optics and the Photon); and the Wannier exciton in a semiconductor, a hydrogen-like bound state of electron and hole with the dielectric constant and effective masses substituted [Wannier:1937], taken up in Semiconductors.]
Exotic atoms
[Reserved: positronium, an electron–positron bound state with reduced mass \(m_{\mathrm{e}}/2\) and no nucleus at all, whose formation and decay were first observed by Deutsch [Deutsch:1951]—the purely leptonic hydrogen, where the annihilation channels test Quantum Electrodynamics and Renormalization; muonium, observed through its Larmor precession [Hughes:1960]; muonic hydrogen [Pohl:2010] [Antognini:2013], whose result reshaped the constants; and antiprotonic and pionic atoms, whose level shifts measure the strong interaction at threshold (Nuclear Forces and Nuclear Structure). Each is the same equation with one substitution, which is what makes hydrogen the reference problem of the treatise.]