Why does the hydrogen energy depend only on the principal quantum number n?

For a pure Coulomb (1/r) potential the energy levels are degenerate in the orbital quantum number l, an accidental degeneracy specific to the inverse-square force; in many-electron atoms screening removes it so energy depends on both n and l.

What is a term symbol?

A term symbol such as ²P₃⁄₂ compactly encodes an atomic state's total spin, total orbital angular momentum, and total angular momentum, summarizing how the electrons' angular momenta couple together.

Atomic Structure and Spectra

Atomic structure and spectra describe how electrons arrange themselves in quantized energy levels around a nucleus and how transitions between those levels produce the characteristic line spectra of the elements.

Definition

Atomic structure and spectra is the study of the bound stationary states of electrons in atoms—their energies, quantum numbers, and spatial distributions—together with the spectral lines emitted or absorbed when electrons make transitions between these states.

Scope

This area covers the quantum-mechanical structure of atoms and the optical spectra it generates: the exact solution of the hydrogen atom, the central-field model and electronic configurations of many-electron atoms, the building of the periodic table from the Pauli principle, and the selection rules and term symbols that govern allowed radiative transitions. It treats energy levels, quantum numbers, and the empirical spectroscopy that first revealed atomic structure, but leaves the finer corrections to the fine- and hyperfine-structure area.

Sub-topics

Core questions

What are the allowed energy levels of an electron bound to a nucleus, and which quantum numbers label them?
How does the Pauli exclusion principle combine with the central-field model to build the electronic configurations of the elements?
Why do atoms emit and absorb light only at discrete wavelengths?
Which transitions are allowed, and what selection rules determine their intensities?

Key concepts

Principal, orbital, and magnetic quantum numbers
Atomic orbitals and electron configurations
Rydberg formula and spectral series
Pauli exclusion principle
Term symbols and LS coupling
Electric-dipole selection rules

Key theories

Bohr model and the quantization of energy: Bohr's 1913 model postulated discrete circular orbits with quantized angular momentum, correctly reproducing the hydrogen spectrum and the Rydberg formula before the full quantum theory existed.
Schrödinger solution of the hydrogen atom: Solving the Schrödinger equation for the Coulomb potential yields exact energy eigenvalues depending only on the principal quantum number and orbital wavefunctions labelled by n, l and m.
Central-field model and the Aufbau principle: Each electron in a many-electron atom is treated as moving in an averaged spherically symmetric potential; filling these orbitals subject to the Pauli principle reproduces the structure of the periodic table.

Clinical relevance

Atomic spectra are the basis of analytical techniques such as atomic absorption and emission spectroscopy used in chemistry and materials analysis, of astronomical spectroscopy used to determine the composition of stars, and of the spectral standards that underpin frequency metrology and atomic clocks.

History

Atomic structure emerged from nineteenth-century spectroscopy, where Balmer and Rydberg found empirical formulas for hydrogen's spectral lines. Bohr's 1913 quantized-orbit model gave the first physical explanation, and the 1925–1926 development of quantum mechanics by Heisenberg and Schrödinger, together with Pauli's exclusion principle, turned atomic spectra into the proving ground of the new theory and explained the periodic table.

Key figures

Niels Bohr
Erwin Schrödinger
Wolfgang Pauli
Johannes Rydberg

Seminal works

bohr1913
bransden2003
foot2005

Frequently asked questions

Why does the hydrogen energy depend only on the principal quantum number n?: For a pure Coulomb (1/r) potential the energy levels are degenerate in the orbital quantum number l, an accidental degeneracy specific to the inverse-square force; in many-electron atoms screening removes it so energy depends on both n and l.
What is a term symbol?: A term symbol such as ²P₃⁄₂ compactly encodes an atomic state's total spin, total orbital angular momentum, and total angular momentum, summarizing how the electrons' angular momenta couple together.