The Hydrogen Atom

Hydrogen is the only atom for which the Schrödinger equation admits an exact closed-form solution. All figures presented here are evaluated at run time from the wavefunctions given in §1.

Schrödinger 1926 · Born 1926 · Pauli 1926 · Condon & Shortley 1935 · Bethe & Salpeter 1957
MODEL CLASS  EXACT ANALYTIC  the rendered field is |ψ|² itself
Z = 1
one proton, one electron
−13.606 eV / n²
bound energies · exact
…
∫|ψ|²dV · live check
…
⟨r⟩ vs analytic · live
…
lyman-α · live
A
active stage
⚠ Bohr 1913 · shown as counterexample
Stage A · |ψ|² sampled in three dimensions

The electron cloud

Each point represents a single position-measurement outcome sampled from |ψₙₗₘ|². Colour encodes the sign of ψ.

Stage

Principal n 1s – 4f
Orbital ℓ
Magnetic m
Angular basis
Bohr 1913 orbit

State · exact quantum numbers

Values are read from the analytic wavefunction and cross-checked by quadrature at run time. Drag to orbit; scroll to zoom.
Cloud samples16,000
Dot size1.0×
Slice contrast γ · Stages B, C0.45

1 · Formulation

The one-electron Coulomb problem is separable and admits an exact closed-form solution. The complete set of expressions evaluated by this page is given below; §3 reports the corresponding verification against the analytic identities.

1.1Hamiltonian and constants of the motion

$$\hat H=-\frac{\hbar^{2}}{2\mu}\nabla^{2}-\frac{e^{2}}{4\pi\varepsilon_{0}\,r},\qquad \nabla^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial r}\!\left(r^{2}\frac{\partial}{\partial r}\right)-\frac{\hat L^{2}}{\hbar^{2}r^{2}}$$
the Laplacian in spherical coordinates, with the angular part expressed through L̂²
$$\hat L^{2}=-\hbar^{2}\!\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right],\qquad \hat L_{z}=-i\hbar\frac{\partial}{\partial\phi}$$
orbital angular momentum operators
$$\big[\hat H,\hat L^{2}\big]=\big[\hat H,\hat L_{z}\big]=\big[\hat L^{2},\hat L_{z}\big]=0 \qquad\Longrightarrow\qquad \psi(r,\theta,\phi)=R(r)\,Y(\theta,\phi)$$
a central potential leaves L̂² and L̂z conserved; the three commuting observables (Ĥ, L̂², L̂z) label the states
$$\hat L^{2}Y_{\ell}^{m}=\hbar^{2}\ell(\ell+1)\,Y_{\ell}^{m},\qquad \hat L_{z}Y_{\ell}^{m}=m\hbar\,Y_{\ell}^{m}$$
the angular eigenvalue equations; ℓ and m are the quantum numbers of the two conserved angular quantities

1.2The separated equations

$$\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right]Y_{\ell}^{m}=-\ell(\ell+1)\,Y_{\ell}^{m}$$
angular equation; single-valuedness in φ and regularity at the poles quantise ℓ and m
$$-\frac{\hbar^{2}}{2\mu}\frac{d^{2}u}{dr^{2}}+\underbrace{\left[-\frac{e^{2}}{4\pi\varepsilon_{0}\,r}+\frac{\hbar^{2}\ell(\ell+1)}{2\mu r^{2}}\right]}_{V_{\rm eff}(r)}u=E\,u, \qquad u_{n\ell}(r)=r\,R_{n\ell}(r)$$
radial equation in the substitution u = rR; the second term of Veff is the centrifugal barrier, which excludes ℓ > 0 states from the origin

1.3Spherical harmonics

$$Y_{\ell}^{m}(\theta,\phi)=(-1)^{m}\sqrt{\frac{2\ell+1}{4\pi}\,\frac{(\ell-m)!}{(\ell+m)!}}\;P_{\ell}^{m}(\cos\theta)\,e^{im\phi}, \qquad Y_{\ell}^{-m}=(-1)^{m}\big(Y_{\ell}^{m}\big)^{\!*}$$
Condon–Shortley phase convention; the (−1)m factor is written explicitly, so Pℓm below carries no phase
$$P_{\ell}^{m}(x)=\big(1-x^{2}\big)^{m/2}\frac{d^{m}}{dx^{m}}P_{\ell}(x), \qquad P_{\ell}(x)=\frac{1}{2^{\ell}\,\ell!}\frac{d^{\ell}}{dx^{\ell}}\big(x^{2}-1\big)^{\ell}$$
associated Legendre function and the Rodrigues form of the Legendre polynomial
$$\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\big(Y_{\ell}^{m}\big)^{\!*}Y_{\ell'}^{m'}\,\sin\theta\,d\theta\,d\phi=\delta_{\ell\ell'}\delta_{mm'}$$
orthonormality on the unit sphere
Spherical harmonics through ℓ = 2, in the convention of Eq. (7)
ℓ, mYℓm(θ, φ)ℓ, mYℓm(θ, φ)
0, 0\(\displaystyle \tfrac{1}{2}\,\pi^{-1/2}\) 2, 0\(\displaystyle \sqrt{\tfrac{5}{16\pi}}\,\big(3\cos^{2}\theta-1\big)\)
1, 0\(\displaystyle \sqrt{\tfrac{3}{4\pi}}\,\cos\theta\) 2, ±1\(\displaystyle \mp\sqrt{\tfrac{15}{8\pi}}\,\sin\theta\cos\theta\;e^{\pm i\phi}\)
1, ±1\(\displaystyle \mp\sqrt{\tfrac{3}{8\pi}}\,\sin\theta\;e^{\pm i\phi}\) 2, ±2\(\displaystyle \sqrt{\tfrac{15}{32\pi}}\,\sin^{2}\theta\;e^{\pm 2i\phi}\)

1.4Radial functions

$$R_{n\ell}(r)=\sqrt{\left(\tfrac{2}{n a_0}\right)^{3}\frac{(n-\ell-1)!}{2n\,(n+\ell)!}}\;e^{-\rho/2}\,\rho^{\ell}\,L_{n-\ell-1}^{\,2\ell+1}(\rho),\qquad \rho=\frac{2r}{n a_0},\qquad a_0=\frac{4\pi\varepsilon_{0}\hbar^{2}}{\mu e^{2}}$$
L denotes the generalized Laguerre polynomial and a₀ the Bohr radius; evaluated directly, without tabulation
$$\int_{0}^{\infty}R_{n\ell}(r)\,R_{n'\ell}(r)\,r^{2}\,dr=\delta_{nn'}$$
radial orthonormality at fixed ℓ
Radial functions through n = 3, in units a0−3/2 with σ = r/a0
n, ℓRnℓ(r)n, ℓRnℓ(r)
1, 0\(\displaystyle 2\,e^{-\sigma}\) 3, 0\(\displaystyle \tfrac{2}{81\sqrt{3}}\,\big(27-18\sigma+2\sigma^{2}\big)\,e^{-\sigma/3}\)
2, 0\(\displaystyle \tfrac{1}{2\sqrt{2}}\,\big(2-\sigma\big)\,e^{-\sigma/2}\) 3, 1\(\displaystyle \tfrac{4}{81\sqrt{6}}\,\big(6-\sigma\big)\,\sigma\,e^{-\sigma/3}\)
2, 1\(\displaystyle \tfrac{1}{2\sqrt{6}}\,\sigma\,e^{-\sigma/2}\) 3, 2\(\displaystyle \tfrac{4}{81\sqrt{30}}\,\sigma^{2}\,e^{-\sigma/3}\)

1.5The complete solution

$$\psi_{n\ell m}(r,\theta,\phi)=R_{n\ell}(r)\,Y_{\ell}^{m}(\theta,\phi),\qquad n=1,2,\dots\quad \ell=0,1,\dots,n-1\quad m=-\ell,\dots,+\ell$$
the three quantum numbers and their admissible ranges
$$E_{n}=-\frac{\mu e^{4}}{2(4\pi\varepsilon_{0})^{2}\hbar^{2}}\,\frac{1}{n^{2}}=-\frac{13.606\ \text{eV}}{n^{2}}, \qquad g_{n}=\sum_{\ell=0}^{n-1}(2\ell+1)=n^{2}$$
the energy depends on n alone — an accidental degeneracy arising from the additional Runge–Lenz symmetry of the Coulomb potential
$$\text{radial nodes}=n-\ell-1,\qquad \text{angular nodes}=\ell,\qquad \text{total nodal surfaces}=n-1$$
the node theorem, counted directly on the stage in Stage B
$$dP=\big|\psi_{n\ell m}\big|^{2}dV=\big|R_{n\ell}(r)\big|^{2}\big|Y_{\ell}^{m}(\theta,\phi)\big|^{2}\,r^{2}\sin\theta\,dr\,d\theta\,d\phi, \qquad P(r)=r^{2}R_{n\ell}^{2}(r)$$
Born's rule: |ψ|² is a probability density, and does not define a trajectory. P(r) is the radial distribution plotted in Stage B

1.6Real angular basis

$$Y_{\ell m}^{\cos}=\sqrt{2}\,\mathrm{Re}\,Y_{\ell}^{m}=\frac{1}{\sqrt{2}}\Big[Y_{\ell}^{m}+(-1)^{m}Y_{\ell}^{-m}\Big], \qquad Y_{\ell m}^{\sin}=\sqrt{2}\,\mathrm{Im}\,Y_{\ell}^{m}=\frac{-i}{\sqrt{2}}\Big[Y_{\ell}^{m}-(-1)^{m}Y_{\ell}^{-m}\Big]$$
m > 0; the two bases span the same subspace and are related by a unitary transformation. Overall sign is conventional
$$p_{z}=Y_{1}^{0},\quad p_{x}=\sqrt2\,\mathrm{Re}\,Y_{1}^{1},\quad p_{y}=\sqrt2\,\mathrm{Im}\,Y_{1}^{1}; \qquad d_{z^{2}}=Y_{2}^{0},\quad d_{xz},d_{yz}\!\leftarrow\! Y_{2}^{\pm1},\quad d_{x^{2}-y^{2}},d_{xy}\!\leftarrow\! Y_{2}^{\pm2}$$
the lobed forms are real combinations; L̂z is not sharp in this basis, and |ψ|² is azimuthally symmetric only in the complex basis

1.7Closed-form moments and selection rules

$$\langle r\rangle=\frac{a_0}{2}\big[3n^{2}-\ell(\ell+1)\big],\qquad \langle r^{2}\rangle=\frac{a_0^{2}n^{2}}{2}\big[5n^{2}+1-3\ell(\ell+1)\big]$$
verified by quadrature in §3 for the 3d state
$$\Big\langle \frac{1}{r}\Big\rangle=\frac{1}{n^{2}a_0},\qquad \Big\langle \frac{1}{r^{2}}\Big\rangle=\frac{1}{n^{3}\big(\ell+\tfrac12\big)a_0^{2}},\qquad \Big\langle \frac{1}{r^{3}}\Big\rangle=\frac{1}{n^{3}\ell\big(\ell+\tfrac12\big)(\ell+1)a_0^{3}}$$
the last requires ℓ > 0; it sets the scale of the spin–orbit coupling omitted in §4
$$\big\langle n'\ell'm'\big|\,\mathbf{r}\,\big|n\ell m\big\rangle\ \ \text{non-vanishing} \qquad\Longleftrightarrow\qquad \Delta\ell=\pm1,\quad \Delta m=0,\pm1$$
electric-dipole selection rules, following from the angular integral; they govern the superpositions available in Stage C

2 · Stages

Select an orbital:

Stagewhat it showscaveat
A · Electron cloud|ψₙₗₘ|² sampled as a three-dimensional point cloud for any (n, ℓ, m) up to n = 4 — 1s through 4f, coloured by the sign of ψpoints are measurement outcomes and do not constitute a trajectory; the lobed forms correspond to one choice of real basis, while the complex m eigenstate is azimuthally symmetric. Both bases are selectable
B · Nodesa signed cross-section of ψ through the x–z plane, plus P(r) = r²R²ℓ angular and n−ℓ−1 radial nodes, counted at run time; the density vanishes exactly on these surfaces
C · Time evolutiona two-state superposition; |Ψ(t)|² oscillates at ω = (Eb − Ea)/ℏa single eigenstate is time-independent; a degenerate pair such as 2s + 2p_z remains stationary
D · Energy ladderthe −13.606/n² levels, the Lyman, Balmer and Paschen series at their vacuum wavelengths, and the omitted corrections drawn to scalethe reduced-mass correction, fine structure, the Lamb shift and hyperfine structure are all absent from this Hamiltonian

Stage A provides an optional overlay of the 1913 Bohr orbit, rₙ = n²a₀, retained as a labelled counterexample. The construction remains the most widely reproduced representation of the atom and is inconsistent with the solution presented here.

3 · Verification computed live

Quantities marked live are evaluated at run time by the routines that generate the figures. Each was additionally verified offline against the corresponding analytic identity to machine precision.

Testthis pageanalytic / measuredstatus
Normalization ∫|ψ|²dV, sampled over (n, ℓ, m)…unity, Eq. (9) and Eq. (11)…
⟨r⟩ for 3d, by quadrature…10.5 a₀, from Eq. (18)…
Radial node count, 4s / 3p / 3d…3 / 1 / 0, from Eq. (14)…
Most probable radius of 1s, from the maximum of r²R²…1.000 a₀, maximum of Eq. (15)…
Lyman-α 2→1 from E_n (point-Coulomb, μ = mₑ)…121.502 nm from Eq. (13); observed 121.568 nm…
Dipole element ⟨1s|z|2p_z⟩ (Stage C amplitude)…128√2⁄243 a₀ = 0.7449 a₀; allowed by Eq. (20)…

4 · Domain of validity

Terms retained in the Hamiltonian and terms omitted

Model class: exact analytic solution of the non-relativistic, spin-free Schrödinger equation for one electron in a fixed point Coulomb field. No quantity is fitted or stylised; the rendered field is |ψ|² itself. The limitations below are not numerical in origin but reflect physics not represented in the Hamiltonian.

Bound-state structure

The energies −13.606/n² eV, the n² degeneracy, the node theorem and all moments ⟨rᵏ⟩ follow in closed form and are verified in §3.

valid · orbital geometry and gross spectrum
Probability density

The point cloud, the nodal surfaces and the superposition dynamics are the Born density and its exact time evolution, observable as intensities rather than as positions.

valid · probability, phase, interference
Finite nuclear mass

The proton is of finite mass. Substituting μ = mₑmₚ/(mₑ+mₚ) for mₑ displaces every level by one part in 1837, corresponding to the 121.502 → 121.568 nm discrepancy reported in §3.

omitted · ~5×10⁻⁴ relative
Spin and relativity

Spin, spin–orbit coupling and the relativistic correction to the kinetic energy split each level at order α²En; at n = 2 the 2p₃/₂–2p₁/₂ separation is 4.5×10⁻⁵ eV (10.9 GHz). Spin is not represented in this Hamiltonian.

omitted · requires Dirac; α² = 5.33×10⁻⁵
Radiative corrections

The Dirac equation retains the degeneracy of 2s₁/₂ and 2p₁/₂. Radiative corrections lift it by 1058 MHz, or 4.4×10⁻⁶ eV.

omitted · requires QED
Hyperfine structure

Coupling to the proton magnetic moment splits the 1s ground state by 1420.406 MHz, the 21 cm transition used to trace neutral hydrogen. Nuclear spin is not represented here.

omitted · 5.9×10⁻⁶ eV · λ = 21.106 cm

Summary. The model comprises a spinless electron, a point proton and no radiation field. The orbital geometry, nodal structure, degeneracies and gross spectrum are exact within it. Fine structure, radiative corrections and the hyperfine transition are drawn to scale in Stage D.

References