Why is only hydrogen solved exactly?

Hydrogen has a single electron, so its Schrödinger equation is a two-body problem reducible to one body in a central potential. With two or more electrons, electron–electron repulsion makes the equation non-separable and only approximate or numerical solutions exist.

What does a quantum defect physically represent?

It quantifies how much a valence electron's orbit penetrates the inner electron shells of an alkali atom. Low-angular-momentum (s, p) orbits penetrate the core and have large defects, while high-l orbits stay outside and behave almost like hydrogen with a near-zero defect.

Hydrogen Atom and Quantum Defects

The hydrogen atom is the only neutral atom solved exactly in quantum mechanics, and its level scheme—modified by a quantum defect—also describes the highly excited Rydberg states of alkali atoms.

Definition

The hydrogen atom is a single electron bound to a proton by the Coulomb force, whose Schrödinger equation is solved exactly; the quantum defect is an empirical correction to the principal quantum number that accounts for the penetration of a valence electron into the ionic core of an alkali atom.

Scope

This topic covers the exact quantum-mechanical solution of one-electron (hydrogenic) atoms: the Coulomb energy eigenvalues, the radial and angular wavefunctions, degeneracy, and the Rydberg spectral series. It extends to alkali and Rydberg atoms, where a penetrating valence electron experiences a screened core and its energy is described by a quantum defect that shifts the effective principal quantum number.

Core questions

What are the exact energy levels and wavefunctions of a one-electron atom?
Why are hydrogen's levels degenerate in the orbital quantum number?
How does a quantum defect modify the Rydberg formula for alkali atoms?
What makes high-lying Rydberg states unusually large and long-lived?

Key concepts

Coulomb potential and reduced mass
Principal, orbital, and magnetic quantum numbers
Radial wavefunctions and nodes
Rydberg constant and spectral series
Quantum defect
Rydberg states and core penetration

Key theories

Exact Coulomb solution: Separating the Schrödinger equation in spherical coordinates for the 1/r potential gives energies E_n = -R/n² and wavefunctions built from associated Laguerre polynomials and spherical harmonics.
Quantum-defect theory: For alkali atoms the valence electron sees a screened nucleus, and its energy follows a modified Rydberg formula E = -R/(n - δ_l)², where the quantum defect δ_l measures core penetration and depends mainly on the orbital quantum number.

Clinical relevance

The hydrogen spectrum sets the values of fundamental constants such as the Rydberg constant and underlies high-precision tests of quantum electrodynamics, while Rydberg atoms—exquisitely sensitive to fields—are used in quantum-information platforms and as sensitive electric-field sensors.

History

The hydrogen spectrum, parameterized by Balmer in 1885 and generalized by Rydberg, was the first quantitative target of atomic theory; Bohr reproduced it in 1913 and Schrödinger derived it exactly in 1926. The quantum defect emerged from spectroscopy of the alkalis, whose lines resemble hydrogen's but are shifted, and was systematized into quantum-defect theory in the twentieth century.

Key figures

Erwin Schrödinger
Johannes Rydberg
Arnold Sommerfeld

Seminal works

bransden2003
gallagher1994

Frequently asked questions

Why is only hydrogen solved exactly?: Hydrogen has a single electron, so its Schrödinger equation is a two-body problem reducible to one body in a central potential. With two or more electrons, electron–electron repulsion makes the equation non-separable and only approximate or numerical solutions exist.
What does a quantum defect physically represent?: It quantifies how much a valence electron's orbit penetrates the inner electron shells of an alkali atom. Low-angular-momentum (s, p) orbits penetrate the core and have large defects, while high-l orbits stay outside and behave almost like hydrogen with a near-zero defect.