Hydrogen Atom: Model and Hamiltonian

Hydrogen is the electron–proton Coulomb problem after separating center-of-mass motion. This reference identifies the ideal nonrelativistic model, its Hamiltonian, and its limits. The Hydrogen Atom teaching page derives the bound-state solution; Hydrogen Spectrum: Formula Reference and Hydrogen Radial Wavefunctions: Formula Reference supply calculation formulas.

Coulomb model and Hamiltonian

For one electron and a point nucleus of charge $+Ze$ and mass $M_N$, define the relative coordinate $\mathbf r=\mathbf r_e-\mathbf r_N$ and reduced mass

$$\mu=\frac{m_eM_N}{m_e+M_N}.$$

In SI units, with $e>0$, the internal Hamiltonian is

$$\hat H_{\rm rel}=\frac{\hat{\mathbf p}^{\,2}}{2\mu}-\frac{Ze^2}{4\pi\epsilon_0r}, \qquad \hat{\mathbf p}=-i\hbar\nabla.$$

Ordinary hydrogen has $Z=1$ and $M_N=m_p$. The total Hamiltonian also contains free center-of-mass kinetic energy $\hat{\mathbf P}^{\,2}/[2(m_e+M_N)]$; quoted atomic levels are internal energies. Replacing $\mu$ by $m_e$ is the infinite-nuclear-mass approximation.

Model feature	Definition or exact target
Degrees of freedom	Electron–nucleus relative coordinate; spin omitted
Hilbert space	$L^2(\mathbb R^3,d^3r)$
Bound-state labels	$n=1,2,\ldots$; $\ell=0,\ldots,n-1$; $m=-\ell,\ldots,\ell$
Bound energies	$E_n=-\mu c^2(Z\alpha_{\rm fs})^2/(2n^2)$
Spatial shell degeneracy	$n^2$; $2n^2$ only when spin is appended without spin-dependent interactions
Energy zero	Separated electron and nucleus; continuum threshold $E=0$
Parity	$(-1)^\ell$
Exact solvability	Bound eigenfunctions and energies; analytic Coulomb continuum functions and phases
Extra symmetry	Conserved Runge–Lenz structure explains the additional Coulomb degeneracy

Here $\alpha_{\rm fs}=e^2/(4\pi\epsilon_0\hbar c)$. The Coulomb operator has its standard self-adjoint realization: bound states are square integrable, regular at the origin, and decay at infinity. In the radial representation $u=rR$, the regular boundary condition is $u(0)=0$. Radial Wavefunctions distinguishes $R$, $u$, and radial probability.

In Hartree atomic units with an infinitely heavy nucleus,

$$\hat H=-\frac12\nabla^2-\frac Zr, \qquad E_n=-\frac{Z^2}{2n^2}.$$

The mass approximation is part of this compact form. Translate Coulomb and unit conventions before combining SI, Gaussian, natural-unit, or atomic-unit formulas.

Physical scope and corrections

Exact solvability applies to instantaneous point-Coulomb interaction without relativity, spin interactions, nuclear structure, radiation coupling, or external fields. It does not make matrix elements, continuum normalization, or precision corrections trivial. Low-lying states require $Z\alpha_{\rm fs}\ll1$ for a controlled nonrelativistic description.

Additional physics	Consequence and explanation
Relativistic kinematics and spin–orbit terms	Fine structure; binding is of order $\mu c^2(Z\alpha_{\rm fs})^2$, fine structure of order $\mu c^2(Z\alpha_{\rm fs})^4$
Radiative corrections	Lamb shifts and other level shifts
Nuclear spin and magnetic moment	Hyperfine structure
Finite nuclear charge radius	Short-distance shifts, especially for states sampling the origin; see hydrogen as an atomic prototype
External electric or magnetic fields	Stark and Zeeman shifts and state mixing
Coupling to radiation	Transition rates, linewidths, and finite excited-state lifetimes
Additional electrons	Screening, exchange, correlation, and antisymmetry; independent exact hydrogen orbitals no longer solve the problem

Excited eigenstates of the static Coulomb Hamiltonian have infinite lifetime. Spontaneous emission requires coupling to electromagnetic modes. Orbitals describe wavefunctions, not electron trajectories; their nodes and choice of real or complex basis are explained in Atomic Orbitals.

Related Coulomb systems

System	Change and limitation
Hydrogenic ions	$Z>1$; relativistic and finite-size effects grow with $Z$
Deuterium and tritium	Nuclear mass and structure change; reduced mass is only the leading isotope correction
Positronium	Equal electron and positron masses; annihilation, spin, and QED effects matter
Muonic atoms	Heavier orbiting particle and smaller radius increase sensitivity to nuclear structure
Coulomb scattering	Positive-energy sector requires continuum and long-range scattering conventions
Alkali valence electrons	Core screening and quantum defects replace exact Coulomb degeneracy
Helium and many-electron atoms	Electron–electron interaction prevents exact independent hydrogen solutions

Neutral helium is not hydrogenic; the one-electron ion $\mathrm{He}^+$ is.

Observables and numerical checks

The characteristic length is $a_Z=4\pi\epsilon_0\hbar^2/(\mu Ze^2)$. These targets make hydrogen useful for checking radial solvers and basis calculations.

Check	Exact point-Coulomb target
Bound energies and spatial degeneracy	$E_n\propto-1/n^2$ and $n^2$
Ground radial probability maximum	$r=a_Z$
Ground radial moments	$\langle r\rangle=3a_Z/2$, $\langle1/r\rangle=1/a_Z$
Virial relation	$2\langle T\rangle=-\langle V\rangle$
Angular eigenvalues	$\hbar^2\ell(\ell+1)$ and $\hbar m$
Radial boundary conditions	$u(0)=0$ and square-integrable decay

A numerical check must also state the origin treatment, outer boundary, angular sector, energy zero, grid or basis, and convergence with resolution and box size. Radial Schrödinger Solvers gives that workflow.

Electric-dipole observables probe transition amplitudes, with $\Delta\ell=\pm1$ and polarization-dependent $\Delta m=0,\pm1$ in the ideal basis; Dipole Transitions gives the tensor-operator statement. Angular momentum and spin determine magnetic response once the relevant terms are included. Ionization requires continuum flux rather than bound-state normalization.

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.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• 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.