Bound-State Spectrum of the Hydrogen Atom

Derivation

Bound-State Spectrum of the Hydrogen Atom

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Statement

For a single electron bound to a point nucleus of charge \( +Ze \) interacting through the Coulomb potential \( V(r) = -\dfrac{Ze^2}{4\pi\varepsilon_0 r} \), the time-independent Schrödinger equation admits square-integrable solutions only at the discrete energies \( E_n = -\dfrac{Z^2 e^4 \mu}{2(4\pi\varepsilon_0)^2 \hbar^2}\,\dfrac{1}{n^2} = -\dfrac{13.6\ \text{eV}}{n^2}\,Z^2 \) for \( n = 1,2,3,\dots \). The bound eigenfunctions factor as \( \psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)\,Y_\ell^m(\theta,\phi) \) with \( R_{n\ell}(r) \propto \left(\tfrac{2r}{na}\right)^{\!\ell} e^{-r/na} L_{n-\ell-1}^{2\ell+1}\!\left(\tfrac{2r}{na}\right) \), and each level \( n \) carries a degeneracy of \( n^2 \) (ignoring spin) from the allowed values \( \ell = 0,\dots,n-1 \) and \( m = -\ell,\dots,+\ell \).

Why it matters

The hydrogen atom is the only realistic three-dimensional bound system whose spectrum can be solved in closed form, and its energy ladder \( E_n \propto -1/n^2 \) reproduced the empirical Rydberg formula that had governed atomic spectroscopy for decades. It is the reference against which every approximate method — perturbation theory, variational bounds, numerical diagonalisation — is calibrated.

The \( n^2 \) degeneracy is not an accident of the potential's shape but a signature of a hidden dynamical symmetry (the conserved Runge–Lenz vector), making hydrogen a gateway to understanding how symmetry organises spectra throughout physics, from the periodic table to the quark model.

Assumptions

Non-relativistic single-particle dynamics.If dropped, the kinetic term \( \hat p^2/2\mu \) is replaced by the Dirac Hamiltonian; fine structure of order \( \alpha^2 E_n \) splits levels of equal \( n \) but different \( j \), and the pure \( n^2 \) degeneracy is broken.

Point nucleus, pure \( 1/r \) potential.A finite nuclear size or any deviation from exact Coulomb form (screening, \( 1/r^{1+\delta} \)) lifts the \( \ell \)-degeneracy, since only the exact Kepler/Coulomb law supports the accidental symmetry.

Reduced mass \( \mu = \dfrac{m_e M}{m_e + M} \) for a two-body atom.Treating the nucleus as infinitely heavy (\( \mu \to m_e \)) shifts every level by the mass ratio \( m_e/M \sim 5\times10^{-4} \), the isotope shift measurable between H and D.

Static, instantaneous Coulomb interaction (no radiation field).Coupling to the quantised electromagnetic field adds the Lamb shift and finite radiative lifetimes; the "stationary" bound states become metastable and the \( 2s\text{–}2p \) degeneracy is removed.

Bound sector only, \( E < 0 \).For \( E \ge 0 \) the spectrum is continuous (scattering states); the quantisation argument below relies specifically on demanding decay as \( r\to\infty \), which only \( E<0 \) permits.

Derivation

We take as given (from central-potential-radial-separation) that \( \psi = R_{n\ell}(r)Y_\ell^m(\theta,\phi) \) and that the radial function obeys the reduced radial equation for \( u(r) = rR(r) \).

\[ -\frac{\hbar^2}{2\mu}\frac{d^2 u}{dr^2} + \left[ -\frac{Ze^2}{4\pi\varepsilon_0 r} + \frac{\hbar^2 \ell(\ell+1)}{2\mu r^2} \right] u = E\,u \]

Starting point: the radial equation with the substitution \( u = rR \), which removes the first-derivative term and turns the problem into a 1D Schrödinger equation on \( r\in[0,\infty) \) with an effective potential (Coulomb plus centrifugal barrier). A

\[ \kappa \equiv \frac{\sqrt{-2\mu E}}{\hbar},\qquad a \equiv \frac{4\pi\varepsilon_0 \hbar^2}{\mu Z e^2},\qquad \rho \equiv \kappa r \]

Define a decay length \( 1/\kappa \) (real for the bound sector \( E<0 \)) and the Bohr length \( a \), then non-dimensionalise the radial coordinate. This strips the equation of dimensional clutter so the eigenvalue emerges as a pure number. A

\[ \frac{d^2 u}{d\rho^2} = \left[ 1 - \frac{\rho_0}{\rho} + \frac{\ell(\ell+1)}{\rho^2} \right] u, \qquad \rho_0 \equiv \frac{Z e^2}{4\pi\varepsilon_0}\frac{\mu}{\hbar^2 \kappa} = \frac{2}{\kappa a} \]

Divide the radial equation by \( E = -\hbar^2\kappa^2/2\mu \) and rewrite in \( \rho \). All physics now sits in the single dimensionless coupling \( \rho_0 \); the eigenvalue condition will fix \( \rho_0 \), hence \( \kappa \), hence \( E \). A

\[ \rho\to\infty:\ \frac{d^2u}{d\rho^2}\approx u \ \Rightarrow\ u\sim e^{-\rho};\qquad \rho\to 0:\ \frac{d^2u}{d\rho^2}\approx \frac{\ell(\ell+1)}{\rho^2}u \ \Rightarrow\ u\sim \rho^{\ell+1} \]

Extract the asymptotic behaviour at both ends. Normalisability forbids the growing solution \( e^{+\rho} \) at infinity and the singular solution \( \rho^{-\ell} \) at the origin, selecting the decaying/regular branches. B

\[ u(\rho) = \rho^{\ell+1} e^{-\rho}\, v(\rho) \]

Peel off both asymptotic factors and let \( v(\rho) \) carry the remaining structure. This is the standard route to a polynomial equation; \( v \) must grow no faster than a power of \( \rho \) so that the \( e^{-\rho} \) factor still controls the tail. B

\[ \rho\frac{d^2 v}{d\rho^2} + 2(\ell+1-\rho)\frac{dv}{d\rho} + \left[\rho_0 - 2(\ell+1)\right]v = 0 \]

Substitute the ansatz of Step 5 into Step 3 and cancel the common factor \( \rho^{\ell}e^{-\rho} \). This is Kummer's confluent hypergeometric / associated-Laguerre equation; recognising it lets us import its known solvability condition. C

\[ v(\rho)=\sum_{k=0}^{\infty} c_k \rho^k,\qquad \frac{c_{k+1}}{c_k} = \frac{2(k+\ell+1)-\rho_0}{(k+1)(k+2\ell+2)} \]

Solve by a Frobenius power series. The two-term recurrence follows by matching coefficients of \( \rho^k \). For large \( k \), \( c_{k+1}/c_k \to 2/k \), which reproduces \( v\sim e^{2\rho} \) and would give \( u\sim e^{+\rho} \) — non-normalisable — unless the series terminates. C

\[ \exists\, k_{\max}\equiv n_r \ge 0:\ c_{n_r+1}=0 \ \Rightarrow\ \rho_0 = 2\,(n_r + \ell + 1) \]

Termination requires the numerator to vanish at some non-negative integer \( n_r \) (the radial quantum number). This is the quantisation condition: only discrete \( \rho_0 \) yield square-integrable states. The surviving \( v \) is a polynomial of degree \( n_r \), namely an associated Laguerre polynomial \( L_{n_r}^{2\ell+1} \). C

\[ n \equiv n_r + \ell + 1 = 1,2,3,\dots \ \Rightarrow\ \rho_0 = 2n \quad\text{and}\quad \kappa = \frac{2}{\rho_0 a} = \frac{1}{na} \]

Define the principal quantum number \( n \) as the integer fixed by termination. Because \( n_r\ge0 \) and \( \ell\ge0 \), we have \( n\ge\ell+1 \), i.e. \( \ell \le n-1 \). Combining \( \rho_0=2/(\kappa a) \) from Step 3 with \( \rho_0=2n \) pins \( \kappa \). B

\[ E = -\frac{\hbar^2\kappa^2}{2\mu} = -\frac{\hbar^2}{2\mu a^2}\frac{1}{n^2} = -\frac{\mu Z^2 e^4}{2(4\pi\varepsilon_0)^2\hbar^2}\frac{1}{n^2} \]

Insert \( \kappa = 1/(na) \) into the definition of \( \kappa \) from Step 2 and substitute \( a = 4\pi\varepsilon_0\hbar^2/\mu Z e^2 \). Symbols only — the numerical Rydberg is evaluated in the Result. A

\[ \sum_{\ell=0}^{n-1}(2\ell+1) = n^2 \]

Count states at fixed \( n \): each \( \ell \) from \( 0 \) to \( n-1 \) supplies \( 2\ell+1 \) values of \( m \), and \( E \) depends on \( n \) alone. The arithmetic-series sum gives the degeneracy \( n^2 \) (times 2 for spin). B

Result

\[ E_n = -\frac{\mu Z^2 e^4}{2(4\pi\varepsilon_0)^2\hbar^2}\,\frac{1}{n^2} = -\frac{13.606\ \text{eV}\,Z^2}{n^2},\qquad g_n = n^2 \]

Reading. The bound energies form a converging ladder accumulating at the ionisation threshold \( E=0 \) as \( n\to\infty \); the ground state (\( n=1 \)) sits \( 13.6\,Z^2 \) eV below threshold. The single integer \( n = n_r + \ell + 1 \) sets the energy, so states of different orbital shape (\( \ell \)) but equal \( n \) are exactly degenerate — the famous "accidental" degeneracy — giving \( n^2 \) orbital states per level. The eigenfunctions are the hydrogenic orbitals, radial nodes numbering \( n_r = n-\ell-1 \).

Units check. \( \dfrac{\mu e^4}{(4\pi\varepsilon_0)^2\hbar^2} \) has units \( \dfrac{\text{kg}\cdot\text{C}^4}{(\text{C}^2\,\text{J}^{-1}\text{m}^{-1})^2\,(\text{J s})^2} = \dfrac{\text{kg}\cdot\text{C}^4\,\text{J}^2\,\text{m}^2}{\text{C}^4\,\text{J}^2\text{s}^2} = \dfrac{\text{kg}\,\text{m}^2}{\text{s}^2} = \text{J} \). Energy, as required; \( n \) and \( Z \) are dimensionless.

Limiting cases

\( n\to\infty \) (Rydberg states): \( E_n\to 0^- \) with level spacing \( E_{n+1}-E_n \approx 2\,\text{Ry}\,Z^2/n^3 \to 0 \); the discrete ladder merges smoothly into the continuum and orbits approach classical Kepler ellipses (correspondence principle).
\( Z=1,\ \mu\to m_e \): recovers ordinary hydrogen with \( \text{Ry} = 13.6057\ \text{eV} \) and Bohr radius \( a_0 = 0.529\ \text{Å} \).
Large \( Z \): energies scale as \( Z^2 \) and orbits shrink as \( 1/Z \); at \( Z\alpha \sim 1 \) (\( Z\sim137 \)) the non-relativistic treatment breaks and Dirac corrections dominate.
\( \ell = n-1 \) (circular states): \( n_r=0 \), the Laguerre polynomial is a constant, \( R\propto r^{n-1}e^{-r/na} \) has no radial nodes — the closest quantum analogue of a classical circular orbit.
\( \hbar\to 0 \): level spacing \( \propto \hbar^{-2}\cdot(\text{fixed}) \) — the discreteness is intrinsically quantum; classically the electron would spiral into the nucleus.

Breaks when

Relativistic / strong-field regime (\( Z\alpha \gtrsim 1 \)): the non-relativistic kinetic energy is invalid; fine structure (\( \sim\alpha^2 \)), the Darwin term and spin–orbit coupling split each \( n \)-level by \( j \), and for \( Z \gtrsim 137 \) the Dirac ground state dives toward the negative-energy continuum. The clean \( -13.6\,Z^2/n^2 \) fails at the \( 10^{-4} \) level even for hydrogen.
Non-Coulombic potential (multi-electron atoms, finite nucleus, QED): screening by other electrons or the Lamb shift removes the accidental \( \ell \)-degeneracy, so states of the same \( n \) but different \( \ell \) no longer coincide (e.g. sodium's \( 3s,3p,3d \) are visibly split). The Runge–Lenz symmetry, and with it the exact \( n^2 \) degeneracy, is destroyed.
Continuum / ionised regime (\( E\ge0 \)): the boundary condition that forced series termination no longer applies; solutions are oscillatory scattering states with a continuous spectrum, and "energy levels" cease to be a meaningful concept.

Failure modes

Confusing \( n \), \( n_r \), and \( \ell \): writing the polynomial degree as \( n \) instead of \( n_r = n-\ell-1 \). The Laguerre index is the radial node count, not the principal quantum number.
Miscounting degeneracy: summing \( \sum_{\ell=0}^{n}(2\ell+1) \) (upper limit \( n \) instead of \( n-1 \)) and getting \( (n+1)^2 \); or forgetting the factor of 2 for spin when it is required.
Using electron mass instead of reduced mass: harmless at 4-significant-figure accuracy for heavy nuclei but wrong for positronium (\( \mu=m_e/2 \), so \( \text{Ry}\to 6.8\ \text{eV} \)) or muonic atoms.
Dropping the \( 4\pi\varepsilon_0 \): mixing Gaussian and SI forms of the Coulomb potential, giving energies off by factors of \( (4\pi\varepsilon_0)^2 \).
Believing the \( \ell \)-degeneracy is generic: assuming any attractive central well gives \( n^2 \) degeneracy. It is special to the exact \( 1/r \) (and the 3D isotropic oscillator); nothing else does it.
Forgetting the \( \rho^{\ell+1} \) prefactor: attempting a bare power series in \( u \) that does not respect the \( r\to0 \) centrifugal behaviour, producing a divergent or non-terminating solution.

Discussion

The energy depends only on \( n = n_r + \ell + 1 \), a combination in which radial and angular excitations trade off freely: promoting one unit of orbital angular momentum while demoting one radial node leaves \( E \) unchanged. This is the hallmark of an "accidental" degeneracy, and its resolution is one of the most elegant results in quantum mechanics. The Coulomb Hamiltonian conserves not only \( \hat{\vec L} \) but also the quantum Laplace–Runge–Lenz vector \( \hat{\vec A} \); together they close into an \( SO(4) \) algebra whose irreducible representations have dimension \( n^2 \). The degeneracy is therefore a symmetry, exactly as the \( (2\ell+1) \)-fold \( m \)-degeneracy is a consequence of rotational \( SO(3) \) symmetry — it only looks accidental if one has not spotted the larger group.

The scale \( a = 4\pi\varepsilon_0\hbar^2/\mu Ze^2 \) and the energy \( \text{Ry} = \hbar^2/2\mu a^2 = \tfrac12 \mu c^2 (Z\alpha)^2 \) can be read off from dimensional analysis alone once one accepts that the only ingredients are \( \hbar \), \( \mu \), and the coupling \( e^2/4\pi\varepsilon_0 \). The fine-structure constant \( \alpha = e^2/4\pi\varepsilon_0\hbar c \approx 1/137 \) organises the hierarchy: the gross structure is \( \tfrac12\mu c^2(Z\alpha)^2 \), fine structure enters at \( (Z\alpha)^4 \), and QED (Lamb shift) at \( \alpha(Z\alpha)^4\ln \). Hydrogen is thus the paradigm for perturbative expansions in a small dimensionless number.

Physically, the ground-state energy is fixed by the competition between the Coulomb attraction, which wants the electron at the origin, and the kinetic penalty of confinement demanded by the uncertainty principle. Minimising \( \langle T\rangle + \langle V\rangle \sim \hbar^2/2\mu r^2 - Ze^2/4\pi\varepsilon_0 r \) over a length scale \( r \) reproduces both \( a \) and \( \text{Ry} \) up to order-unity factors, showing that the atom's very stability is a quantum effect — the answer to why matter does not collapse.

The bound spectrum is only half the story. The same radial equation for \( E>0 \) yields Coulomb scattering wavefunctions with a logarithmically distorted phase (the long range of \( 1/r \) means no ordinary asymptotic plane wave), and analytically continuing \( n \to \) complex values locates the bound states as poles of the scattering \( S \)-matrix at \( E_n \). The discrete ladder and the continuum are two faces of one analytic object; the same \( SO(4) \) that organises the bound states enlarges to \( SO(4,1) \) or \( SO(4,2) \) dynamical (spectrum-generating) algebras that act on the entire spectrum at once — a perspective that generalises to the hydrogen atom's surprising appearances in supersymmetric quantum mechanics and in the AdS/CFT-adjacent literature.

Common misconceptions. The formula \( E_n=-13.6/n^2 \) eV is often quoted as "the" hydrogen spectrum, but it is the leading term of an expansion, degenerate in \( \ell \) and \( j \) only at this order; real hydrogen shows fine structure, hyperfine structure and the Lamb shift. Equally, students often think the \( m \)-degeneracy and the \( \ell \)-degeneracy have the same origin — they do not: the former is universal to any central potential, the latter is unique to \( 1/r \).

Worked examples

Example 1 — Wavelength of the Balmer-alpha line (\( n=3\to n=2 \) in hydrogen).

\[ \Delta E = E_2 - E_3 = -13.606\,\text{eV}\left(\frac{1}{2^2}-\frac{1}{3^2}\right) \]

Photon energy equals the energy difference between the initial and final bound levels (energy conservation). Symbols first. A

\[ \Delta E = 13.606\,\text{eV}\left(\frac{1}{4}-\frac{1}{9}\right) = 13.606\,\text{eV}\times\frac{5}{36} = 1.889\ \text{eV} \]

Insert numbers; \( \tfrac14-\tfrac19 = \tfrac{9-4}{36}=\tfrac{5}{36} \). A

\[ \lambda = \frac{hc}{\Delta E} = \frac{1239.84\ \text{eV·nm}}{1.889\ \text{eV}} \]

Photon wavelength from \( E=hc/\lambda \); use the convenient constant \( hc = 1239.84\ \text{eV·nm} \). A

\[ \lambda \approx 656.3\ \text{nm} \]

Reading. This is the red H\( \alpha \) line, the brightest Balmer feature and the characteristic glow of ionised-hydrogen nebulae. Units check. eV·nm / eV = nm.

Example 2 — Ground-state binding energy of a muonic lead ion (\( \mu^- \) orbiting \( Z=82 \), point-nucleus estimate).

\[ E_1 = -13.606\,\text{eV}\;Z^2\;\frac{\mu_{\text{red}}}{m_e} \]

The Rydberg scales as \( Z^2 \) and linearly with the reduced mass; replace the electron by a muon. Symbols first. B

\[ \mu_{\text{red}} = \frac{m_\mu M}{m_\mu + M}\approx m_\mu = 206.77\,m_e \quad(\text{since } M_{\text{Pb}}\gg m_\mu) \]

The lead nucleus (\( \sim 2\times10^5\,m_e \)) is far heavier than the muon, so the reduced mass is essentially the muon mass. B

\[ E_1 = -13.606\,\text{eV}\times 82^2 \times 206.77 = -13.606\times 6724 \times 206.77\ \text{eV} \]

Insert \( Z=82 \Rightarrow Z^2=6724 \) and the mass ratio. A

\[ E_1 \approx -1.89\times10^{7}\ \text{eV} \approx -18.9\ \text{MeV} \]

Reading. The point-nucleus estimate is enormous — tens of MeV — which is exactly why muonic atoms are so sensitive to nuclear structure: the muon's Bohr radius \( a = a_0/(Z\,206.77) \approx 3\ \text{fm} \) is comparable to the nuclear radius, so the real binding is much reduced and relativistic Dirac corrections are essential. The naive formula flags its own breakdown. Units check. eV × (dimensionless) × (dimensionless) = eV.

Problems

Compute the ionisation energy of singly-ionised helium (\( \text{He}^+ \), \( Z=2 \)) from its ground state, treating the nucleus as infinitely heavy.
Solution
\( E_1 = -13.606\,Z^2/n^2 = -13.606\times4/1 = -54.42\ \text{eV} \). The ionisation energy is \( +54.4\ \text{eV} \). (The measured value is 54.4 eV; agreement is excellent because He\(^+\) is genuinely one-electron.)
How many bound states (counting \( \ell,m \), ignoring spin) exist with principal quantum number \( n=4 \)? List the \( (\ell,m) \) multiplicities.
Solution
Degeneracy \( = n^2 = 16 \). Breakdown: \( \ell=0 \) gives 1 (\( 4s \)); \( \ell=1 \) gives 3 (\( 4p \)); \( \ell=2 \) gives 5 (\( 4d \)); \( \ell=3 \) gives 7 (\( 4f \)). Sum \( 1+3+5+7 = 16 \). With spin, 32.
The Lyman-alpha transition (\( n=2\to1 \)) in hydrogen. Find its photon energy and wavelength, and state the spectral region.
Solution
\( \Delta E = 13.606(1 - \tfrac14) = 13.606\times\tfrac34 = 10.20\ \text{eV} \). \( \lambda = 1239.84/10.20 = 121.6\ \text{nm} \). This lies in the ultraviolet (vacuum-UV); it is the strongest hydrogen line and a key diagnostic of the interstellar medium.
Positronium is an \( e^-e^+ \) bound pair. Using the reduced-mass scaling, find its ground-state energy and Bohr radius relative to hydrogen.
Solution
Reduced mass \( \mu = m_e m_e/(m_e+m_e) = m_e/2 \). Energy scales linearly with \( \mu \): \( E_1 = -13.606/2 = -6.80\ \text{eV} \). The Bohr radius scales as \( 1/\mu \): \( a = 2a_0 = 1.06\ \text{Å} \). (Real positronium also has fine/hyperfine and annihilation effects, but this is the gross structure.)
Show that the level spacing \( E_{n+1}-E_n \) for large \( n \) scales as \( n^{-3} \), and estimate the spacing between the \( n=100 \) and \( n=101 \) Rydberg levels of hydrogen.
Solution
\( E_{n+1}-E_n = -13.606\left(\tfrac{1}{(n+1)^2}-\tfrac{1}{n^2}\right) = 13.606\,\dfrac{(n+1)^2-n^2}{n^2(n+1)^2} = 13.606\,\dfrac{2n+1}{n^2(n+1)^2} \). For large \( n \) this \( \approx 13.606\cdot\dfrac{2n}{n^4} = \dfrac{27.2}{n^3} \), confirming \( n^{-3} \). At \( n=100 \): \( \Delta E \approx 27.2/10^{6} = 2.72\times10^{-5}\ \text{eV} = 27.2\ \mu\text{eV} \) (corresponding to a microwave/far-IR transition near \( \lambda\approx4.6\,\text{cm} \)).