Hydrogen Atom

The hydrogen atom is the nonrelativistic electron–proton Coulomb problem after separation of center-of-mass and relative motion. Its internal Hamiltonian is exactly soluble and supplies the canonical quantum numbers, orbitals, radial functions, degeneracies, and energy scale of one-electron atoms.

The exactness belongs to a sharply defined model: point charges, instantaneous Coulomb interaction, no relativistic terms, no coupling to the quantized electromagnetic field, no nuclear structure, and no external fields. Precision hydrogen requires a hierarchy of corrections beyond this baseline.

Model at a Glance

Field	Standard internal hydrogen model
Degrees of freedom	Electron–proton relative coordinate $\mathbf r$
Hilbert space	$L^2(\mathbb R^3,d^3r)$, with spin omitted
Hamiltonian	$\hat{\mathbf p}^{\,2}/(2\mu)-e^2/(4\pi\epsilon_0r)$
Reduced mass	$\mu=m_em_p/(m_e+m_p)$
Bound-state labels	$n,\ell,m$
Bound energies	$E_n\propto-1/n^2$
Bound spatial degeneracy	$n^2$ in the ideal Coulomb model
Continuum threshold	$E=0$
Solvability	Exact separation into spherical harmonics and associated Laguerre functions
Key special symmetry	Conserved Runge–Lenz structure in addition to rotations
Canonical home	Hydrogen Atom

For a hydrogenic ion with nuclear charge $+Ze$, replace the Coulomb coupling by $Z e^2$. The Hydrogen Atom Hamiltonian is the compact operator card; this page identifies the model and its physical scope.

Physical Setup and Two-Body Reduction

Before separating coordinates, the spinless nonrelativistic Hamiltonian is

$$\hat H_{\mathrm{tot}} =\frac{\hat{\mathbf p}_e^{\,2}}{2m_e} +\frac{\hat{\mathbf p}_N^{\,2}}{2m_N} -\frac{Ze^2} {4\pi\epsilon_0 \lvert\mathbf r_e-\mathbf r_N\rvert}.$$

Define total and reduced masses

$$M=m_e+m_N, \qquad \mu=\frac{m_em_N}{m_e+m_N},$$

and use center-of-mass and relative coordinates. The Hamiltonian separates:

$$\hat H_{\mathrm{tot}} =\frac{\hat{\mathbf P}^{\,2}}{2M} +\hat H_{\mathrm{rel}},$$ $$\hat H_{\mathrm{rel}} =\frac{\hat{\mathbf p}^{\,2}}{2\mu} -\frac{Ze^2}{4\pi\epsilon_0r}.$$

The free center-of-mass plane wave does not affect the internal spectrum. The model card concerns $\hat H_{\mathrm{rel}}$. For ordinary hydrogen, $Z=1$ and $m_N=m_p$.

Using $m_e$ instead of $\mu$ is the infinite-nuclear-mass approximation. It is often adequate for qualitative work but should not be silently mixed with reduced-mass spectroscopic formulas.

Hilbert Space and Domain

The internal spinless Hilbert space is

$$\mathcal H_{\mathrm{rel}} =L^2(\mathbb R^3,d^3r).$$

The Coulomb singularity is part of a well-defined self-adjoint Schrödinger operator on its standard domain. Physical bound states are square integrable, regular at the origin in the full three-dimensional sense, and decay at infinity.

After angular separation,

$$\psi_{n\ell m}(r,\theta,\phi) =R_{n\ell}(r)Y_\ell^m(\theta,\phi).$$

The radial normalization is

$$\int_0^\infty \lvert R_{n\ell}(r)\rvert^2r^2\,dr=1.$$

If $u_{n\ell}(r)=rR_{n\ell}(r)$, then

$$\int_0^\infty \lvert u_{n\ell}(r)\rvert^2\,dr=1,$$

and the regular radial boundary condition is $u(0)=0$. Confusing $R$, $u$, and the radial probability density is one of the most common errors in this model.

Parameters and Natural Scales

Symbol	Meaning	Role
$m_e$	Electron mass	Sets the infinite-nuclear-mass scale
$m_N$	Nuclear mass	Enters the reduced mass
$\mu$	Electron–nucleus reduced mass	Controls orbital length and binding energy
$Z$	Nuclear charge number	$Z=1$ for hydrogen
$e$	Elementary charge magnitude	Coulomb coupling
$\epsilon_0$	Vacuum permittivity	SI-unit convention
$\alpha_{\mathrm{fs}}$	Fine-structure constant	Nonrelativistic expansion parameter through $Z\alpha_{\mathrm{fs}}$

Define the reduced-mass Bohr radius

$$a_\mu =\frac{4\pi\epsilon_0\hbar^2}{\mu e^2}$$

and the hydrogenic length scale

$$a_Z=\frac{a_\mu}{Z}.$$

The corresponding binding-energy scale is

$$E_{\mathrm C} =\frac{\hbar^2}{2\mu a_\mu^2} =\frac{\mu c^2\alpha_{\mathrm{fs}}^2}{2}.$$

For a hydrogenic ion, the leading binding scale grows as $\mu Z^2$, while the orbital scale shrinks as $1/(\mu Z)$.

In Hartree atomic units with an infinitely heavy nucleus, the internal Hamiltonian is

$$\hat H =-\frac12\nabla^2-\frac{Z}{r},$$

and energies are

$$E_n=-\frac{Z^2}{2n^2}$$

in hartrees. Do not combine this compact form with SI parameters without translating units.

Solvability and Quantum Numbers

Rotational symmetry permits simultaneous eigenstates of

$$\hat H, \qquad \hat L^2, \qquad \hat L_z.$$

The bound-state quantum numbers obey

$$n=1,2,3,\ldots,$$ $$\ell=0,1,\ldots,n-1,$$ $$m=-\ell,-\ell+1,\ldots,\ell.$$

The angular functions are Spherical Harmonics, and the radial polynomial factors use Laguerre Polynomials. The complete separation and normalization are owned by Radial Schrödinger Equation and Radial Wavefunctions.

Exact solvability includes:

• the complete discrete bound spectrum;
• normalized bound eigenfunctions;
• analytic continuum Coulomb functions and scattering phases;
• separation into radial and angular sectors;
• algebraic symmetry methods for the bound spectrum.

Closed form does not mean every observable is a one-line expression. Continuum normalization, matrix elements, sums over intermediate states, and precision corrections can remain technically demanding.

Bound Spectrum and Degeneracy

The nonrelativistic hydrogenic bound energies are

$$E_n =-\frac{\mu Z^2e^4} {2(4\pi\epsilon_0)^2\hbar^2} \frac{1}{n^2}, \qquad n=1,2,3,\ldots.$$

Equivalently,

$$E_n =-\frac{\mu c^2(Z\alpha_{\mathrm{fs}})^2}{2n^2}.$$

The zero of energy is the separated electron–nucleus threshold. Bound states have $E<0$, while the continuum has $E>0$. The bound levels accumulate at zero from below.

For fixed $n$, the number of spatial states is

$$g_n =\sum_{\ell=0}^{n-1}(2\ell+1) =n^2.$$

Rotational symmetry explains degeneracy in $m$ for any central potential. The additional independence from $\ell$ is special to the $1/r$ potential and is tied to the conserved quantum Runge–Lenz structure. Degeneracy of the Hydrogen Atom develops this hidden symmetry.

Appending electron spin while omitting all spin-dependent terms doubles the spatial count to $2n^2$. Real fine, hyperfine, radiative, nuclear-size, and external-field terms split parts of this ideal degeneracy.

Eigenstates and Orbitals

The normalized ground state is

$$\psi_{100}(\mathbf r) =\frac{1}{\sqrt{\pi a_Z^3}} \exp\left(-\frac{r}{a_Z}\right).$$

Its probability density is largest at the origin, but the probability per unit radius is

$$P_{100}(r) =4\frac{r^2}{a_Z^3} \exp\left(-\frac{2r}{a_Z}\right),$$

which is maximal at $r=a_Z$.

For a general bound state:

• the number of radial nodes is $n-\ell-1$;
• angular nodes are determined by $Y_\ell^m$;
• parity is $(-1)^\ell$;
• $s,p,d,\ldots$ label $\ell=0,1,2,\ldots$;
• complex $m$ eigenstates and real orbital combinations are different bases within degenerate subspaces.

An orbital is a stationary one-electron wavefunction, not a path followed by an electron. Atomic Orbitals owns the basis, node, and visualization conventions.

Key Observables

The most common observables and diagnostics are:

Quantity	What it probes
$\hat H$	Internal binding energy
$\hat L^2$ and $\hat L_z$	Orbital angular momentum labels
Parity	Spatial inversion symmetry and selection structure
$r$ and radial projectors	Radial size and probability
Electric dipole $\propto\hat{\mathbf r}$	Leading optical transition amplitudes
$\hat{\mathbf L}$ and spin magnetic moment	Zeeman response after spin is included
Continuum flux	Ionization and Coulomb scattering

For the ground state,

$$\langle r\rangle_{100} =\frac32a_Z, \qquad \left\langle\frac{1}{r}\right\rangle_{100} =\frac{1}{a_Z}.$$

The Coulomb virial theorem gives

$$2\langle T\rangle=-\langle V\rangle,$$

so for any stationary bound state,

$$\langle T\rangle=-E_n, \qquad \langle V\rangle=2E_n.$$

Electric-dipole matrix elements obey $\Delta\ell=\pm1$ and polarization-dependent $\Delta m=0,\pm1$ in the ideal basis. The full tensor-operator statement belongs at Dipole Transitions.

What the Model Teaches

Hydrogen is the canonical model for:

• exact two-body reduction and reduced mass;
• central-potential separation in three dimensions;
• spherical harmonics and radial special functions;
• discrete and continuum sectors in one Hamiltonian;
• principal, orbital, and magnetic quantum numbers;
• radial probability measures and nodal structure;
• ordinary rotational degeneracy versus hidden Coulomb degeneracy;
• selection rules and spectroscopic line organization;
• symmetry breaking by controlled corrections;
• radial-solver and basis-set benchmarks.

It is also the bridge from one-particle quantum mechanics to atomic structure. The labels survive in approximate form for more complex atoms even when the exact one-electron Coulomb solution does not.

Scope and Correction Hierarchy

The exact card Hamiltonian is not a precision theory of real hydrogen.

Added physics	What changes	Canonical route
Relativistic kinematics and spin-orbit terms	Fine-structure splitting and corrected energies	Fine Structure
Quantized-field radiative corrections	Lamb shift and radiative level shifts	Lamb Shift Overview
Nuclear magnetic moment and electron–nucleus spin coupling	Hyperfine multiplets	Hyperfine Structure
Finite nuclear charge radius	Short-distance level shifts, strongest for states sampling the origin	Hydrogen prototype and precision atomic pages
External electric field	Stark shifts and mixed parity	Stark Effect in Atoms
External magnetic field	Zeeman splitting and angular-momentum recoupling	Zeeman Effect in Atoms
Coupling to radiation	Transition rates, linewidths, and finite excited-state lifetimes	Atomic and light–matter interaction pages
Additional electrons	Screening, exchange, correlation, and antisymmetry	Many-electron atomic structure methods

The nonrelativistic expansion requires $Z\alpha_{\mathrm{fs}}\ll1$. The leading binding scale is of order $\mu c^2(Z\alpha_{\mathrm{fs}})^2$, while relativistic fine-structure corrections enter at a smaller absolute scale of order $\mu c^2(Z\alpha_{\mathrm{fs}})^4$. High-$Z$ one-electron ions require increasingly relativistic and finite-nuclear-size treatment.

The static Coulomb Hamiltonian gives infinitely lived excited eigenstates. Spontaneous emission appears only after coupling the atom to quantized electromagnetic modes; it is not hidden inside the Schrödinger eigenvalue problem.

Variants and Related Models

Model	Change from ordinary hydrogen	Main caution
Hydrogenic Ion	Nuclear charge $Z>1$	Relativistic and nuclear-size effects grow with $Z$
Deuterium or tritium	Different nuclear mass and structure	Reduced-mass and isotope shifts are not the whole precision correction
Positronium	Equal electron and positron masses	Annihilation, spin, and QED effects are essential
Muonic atom	Electron replaced by a muon	Much smaller orbital scale strongly probes nuclear size
Coulomb scattering	Positive-energy sector	Requires continuum and long-range scattering conventions
Alkali valence electron	Effective central field with a core	Quantum defects replace exact hydrogenic degeneracy
Helium and many-electron atoms	Electron–electron interaction added	No separation into independent exact hydrogen orbitals

Hydrogen as Atomic Prototype organizes how the ideal labels and correction hierarchy are used in atomic physics.

Computational Benchmarks

Hydrogen is a demanding numerical benchmark because of the Coulomb singularity, the half-line radial domain, and the continuum threshold.

Check	Exact target
Bound energies	$E_n\propto-1/n^2$
Spatial shell degeneracy	$n^2$
Ground radial maximum	$r=a_Z$
Ground mean radius	$3a_Z/2$
Virial theorem	$2\langle T\rangle=-\langle V\rangle$
Angular eigenvalues	$\hbar^2\ell(\ell+1)$ and $\hbar m$
Radial boundary	$u(0)=0$ and square-integrable decay for bound states

A radial solver should state the origin treatment, outer boundary, grid or basis, angular sector, energy zero, and convergence with resolution and box size. Radial Schrödinger Solvers gives the computational route.

Common Mistakes

• Replacing the reduced mass by $m_e$ without declaring the fixed-nucleus approximation.
• Forgetting that the card Hamiltonian describes relative motion after center-of-mass separation.
• Mixing SI, Gaussian, natural, and atomic-unit Coulomb conventions.
• Using the $13.6\,\mathrm{eV}$ hydrogen scale for a hydrogenic ion without the $Z^2$ and reduced-mass factors.
• Treating the $n^2$ spatial degeneracy as generic to all central potentials.
• Assuming the ideal degeneracy survives fine, hyperfine, radiative, nuclear-size, or field corrections.
• Confusing $R(r)$, $u(r)=rR(r)$, $\lvert R(r)\rvert^2$, and the radial probability $\lvert R(r)\rvert^2r^2dr$.
• Drawing orbitals as electron trajectories or rigid material surfaces.
• Forgetting the positive-energy Coulomb continuum.
• Using nonrelativistic hydrogenic scaling at large $Z$ without checking $Z\alpha_{\mathrm{fs}}$.
• Treating stationary excited states of the isolated Coulomb Hamiltonian as a theory of radiative lifetimes.
• Applying the one-electron solution directly to many-electron atoms.

Exercises

1. Center-of-mass reduction

Starting from the two-particle kinetic energy, show why the internal kinetic term contains the reduced mass.

Solution

Define

$$\mathbf R =\frac{m_e\mathbf r_e+m_N\mathbf r_N}{M}, \qquad \mathbf r=\mathbf r_e-\mathbf r_N,$$

where $M=m_e+m_N$. The conjugate total and relative momenta can be chosen so that the canonical transformation gives

$$\frac{\hat{\mathbf p}_e^{\,2}}{2m_e} +\frac{\hat{\mathbf p}_N^{\,2}}{2m_N} =\frac{\hat{\mathbf P}^{\,2}}{2M} +\frac{\hat{\mathbf p}^{\,2}}{2\mu},$$

with

$$\frac{1}{\mu} =\frac{1}{m_e}+\frac{1}{m_N}.$$

Therefore

$$\mu=\frac{m_em_N}{m_e+m_N}.$$

Because the Coulomb potential depends only on $\mathbf r$, center-of-mass and internal motion separate.

2. Hydrogenic scaling

Ignoring the small reduced-mass difference, compare the $n=1$ and $n=2$ energies of $\mathrm{He}^+$ with those of hydrogen. Also compare the hydrogenic length scales.

Solution

For a one-electron ion,

$$E_n(Z)=Z^2E_n(Z=1).$$

Thus the $\mathrm{He}^+$ ground state with $Z=2$ is four times as deeply bound as the hydrogen ground state:

$$E_1(\mathrm{He}^+) \approx-4(13.6\,\mathrm{eV}).$$

For $n=2$,

$$E_2(\mathrm{He}^+) =-\frac{4(13.6\,\mathrm{eV})}{4} \approx-13.6\,\mathrm{eV}.$$

The basic Coulomb length is

$$a_Z=\frac{a_\mu}{Z},$$

so the $\mathrm{He}^+$ orbital scale at fixed quantum numbers is half the corresponding hydrogenic scale, before reduced-mass and relativistic corrections.

3. Shell degeneracy and its scope

Show that the ideal spatial degeneracy at fixed $n$ is $n^2$. Explain why this is not the observed degeneracy of precision hydrogen.

Solution

For each $\ell=0,\ldots,n-1$, there are $2\ell+1$ magnetic quantum numbers. Therefore

$$\begin{aligned} g_n &=\sum_{\ell=0}^{n-1}(2\ell+1)\\ &=n(n-1)+n\\ &=n^2. \end{aligned}$$

This counts spatial states in the spinless nonrelativistic point-Coulomb model. Electron spin adds a trivial factor of two only if spin-dependent interactions are still omitted. Fine structure, the Lamb shift, hyperfine coupling, finite nuclear size, and external fields resolve or reorganize parts of the ideal degeneracy.

Canonical Links

• Hydrogen Atom is the canonical nonrelativistic solution.
• Hydrogen Atom Hamiltonian is the operator card.
• Hydrogen Spectrum is the compact spectrum card.
• Coulomb Potential fixes coupling and unit conventions.
• Radial Wavefunctions owns normalization, nodes, and radial moments.
• Continuum States of the Coulomb Problem covers ionized states.
• Hydrogen Atom Angular Structure organizes angular quantum numbers and multiplets.
• Hydrogen as Atomic Prototype connects the model to real atomic structure and spectroscopy.

References

• H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, 2 vols., Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.