Hydrogen Atom

The hydrogen atom is the exactly solvable quantum problem of a single electron bound to a proton by the Coulomb force; its solution reproduces the observed spectrum, defines the atomic orbitals, and serves as the template for understanding all atoms.

Definition

The hydrogen atom is the bound system of an electron and a proton interacting through the Coulomb potential, whose stationary states are labeled by principal, orbital, and magnetic quantum numbers and whose energies, in the simplest treatment, depend only on the principal quantum number.

Scope

The topic covers the separation of the Coulomb central-force problem into radial and angular parts, the spherical harmonics for the angular factor and the associated Laguerre functions for the radial factor, the discrete bound-state energies depending only on the principal quantum number, the resulting orbitals and their quantum numbers, the accidental degeneracy and its hidden symmetry, and the fine and hyperfine corrections.

Core questions

• How does separating the Coulomb problem yield the hydrogen energy levels and orbitals?
• Why do hydrogen energies depend, to leading order, only on the principal quantum number?
• What is the accidental degeneracy and what symmetry explains it?
• How do spin and relativistic effects refine the spectrum into fine and hyperfine structure?

Key concepts

• Coulomb potential
• principal quantum number
• atomic orbitals
• Bohr radius
• accidental degeneracy
• fine and hyperfine structure

Key theories

Coulomb bound states

Solving the radial Schrodinger equation for the Coulomb potential gives normalizable solutions only at discrete energies that scale inversely with the square of the principal quantum number, reproducing the Balmer and Lyman series and Bohr's energy formula from first principles.

Accidental degeneracy and hidden symmetry

Hydrogen levels with the same principal quantum number but different orbital angular momentum share the same energy, an accidental degeneracy explained by a hidden symmetry associated with the conserved Runge-Lenz vector unique to the inverse-square force.

Clinical relevance

The hydrogen solution is the foundation of atomic physics and chemistry: it defines the orbital language used for all elements, explains atomic spectra and the Rydberg formula, and its fine and hyperfine structure underlies precision spectroscopy, atomic clocks, and the 21-centimeter line used in radio astronomy.

History

Bohr's 1913 model first gave the hydrogen spectrum, and Pauli derived it algebraically using the Runge-Lenz vector in 1926, the same year Schrodinger obtained the full wave-mechanical solution; Sommerfeld and later Dirac added the fine structure, and Lamb's measurements drove quantum electrodynamics.

Key figures

• Niels Bohr
• Erwin Schrodinger
• Wolfgang Pauli
• Arnold Sommerfeld

Seminal works

• bethesalpeter1957
• griffiths2018

Frequently asked questions

Why is the hydrogen atom so important in quantum mechanics?

It is the only neutral atom that can be solved exactly, so it provides the reference solution against which approximations for all other atoms are built, and it confirmed quantum mechanics by reproducing the observed spectrum from first principles.

Why do hydrogen energy levels depend only on the principal quantum number?

The inverse-square Coulomb force has an extra conserved quantity, the Runge-Lenz vector, whose associated hidden symmetry makes states of different orbital angular momentum but the same principal quantum number degenerate, a coincidence broken by relativistic and spin corrections.