Sommerfeld quantization rules

E234760

old quantum theory physical theory quantization rule

Sommerfeld quantization rules are an early quantum theory refinement of Bohr’s model that quantize electron motion in elliptical orbits using action integrals, helping to explain fine-structure details in atomic spectra.

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All labels observed (9)

Label	Occurrences
Bohr–Sommerfeld quantization	2
Bohr quantization condition	1
Bohr–Sommerfeld model	1
Bohr–Sommerfeld quantization rules	1
Old Quantum Theory	1
Sommerfeld model of the atom	1
Sommerfeld quantization rules canonical	1
Sommerfeld–Wilson quantization	1
old quantum theory	1

How this entity was disambiguated

This entity first appeared as the object of triple T2093718 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Sommerfeld quantization rules
Context triple: [Arnold Sommerfeld, knownFor, Sommerfeld quantization rules]

A. Vorlesungen über Atommechanik
Vorlesungen über Atommechanik is a foundational early 20th-century textbook on quantum theory and atomic mechanics written by physicist Max Born.
B. Rayleigh–Schrödinger perturbation theory
Rayleigh–Schrödinger perturbation theory is a fundamental method in quantum mechanics for approximating the energies and states of a system by treating interactions as small corrections to an exactly solvable problem.
C. Quantum Mechanics (with Philip M. Morse)
"Quantum Mechanics (with Philip M. Morse)" is a foundational early 20th-century textbook on quantum theory co-authored by physicists Edward Condon and Philip M. Morse.
D. The Principles of Quantum Mechanics
The Principles of Quantum Mechanics is a foundational 1930 textbook by Paul Dirac that rigorously formalized quantum theory and introduced key concepts such as bra–ket notation.
E. Theory of Atomic Spectra (1935)
Theory of Atomic Spectra (1935) is a foundational physics monograph by Edward Condon that systematically develops the quantum-mechanical theory underlying atomic spectral lines and their structure.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Sommerfeld quantization rules
Target entity description: Sommerfeld quantization rules are an early quantum theory refinement of Bohr’s model that quantize electron motion in elliptical orbits using action integrals, helping to explain fine-structure details in atomic spectra.

A. Vorlesungen über Atommechanik
Vorlesungen über Atommechanik is a foundational early 20th-century textbook on quantum theory and atomic mechanics written by physicist Max Born.
B. Rayleigh–Schrödinger perturbation theory
Rayleigh–Schrödinger perturbation theory is a fundamental method in quantum mechanics for approximating the energies and states of a system by treating interactions as small corrections to an exactly solvable problem.
C. Quantum Mechanics (with Philip M. Morse)
"Quantum Mechanics (with Philip M. Morse)" is a foundational early 20th-century textbook on quantum theory co-authored by physicists Edward Condon and Philip M. Morse.
D. The Principles of Quantum Mechanics
The Principles of Quantum Mechanics is a foundational 1930 textbook by Paul Dirac that rigorously formalized quantum theory and introduced key concepts such as bra–ket notation.
E. Theory of Atomic Spectra (1935)
Theory of Atomic Spectra (1935) is a foundational physics monograph by Edward Condon that systematically develops the quantum-mechanical theory underlying atomic spectral lines and their structure.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	old quantum theory ⓘ physical theory ⓘ quantization rule ⓘ
aimsTo	quantize multi-dimensional periodic motion ⓘ
alternativeName	Bohr–Sommerfeld quantization rules ⓘ linked to: Sommerfeld quantization rules
appliesTo	bound systems ⓘ electron orbits ⓘ elliptical orbits ⓘ integrable Hamiltonian systems ⓘ
assumes	classical trajectories ⓘ elliptical Kepler orbits for electrons ⓘ
basedOn	action integral quantization ⓘ
compatibleWith	Bohr–Sommerfeld quantization ⓘ linked to: Sommerfeld quantization rules
explains	relativistic corrections to hydrogen energy levels (partially) ⓘ
extends	Bohr quantization condition ⓘ linked to: Sommerfeld quantization rules
field	atomic physics ⓘ quantum physics ⓘ theoretical physics ⓘ
formulatedBy	Arnold Sommerfeld ⓘ
helpsExplain	Zeeman effect ⓘ anomalies in hydrogen spectral lines ⓘ fine-structure splitting in atomic spectra ⓘ
historicalContext	old quantum theory between classical mechanics and modern quantum mechanics ⓘ
historicalPeriod	early 20th century ⓘ
influenced	development of quantum mechanics ⓘ
involves	action-angle variables ⓘ closed phase-space trajectories ⓘ
mathematicalDomain	Hamiltonian mechanics ⓘ canonical transformations ⓘ
motivatedBy	fine structure of hydrogen spectrum ⓘ
namedAfter	Arnold Sommerfeld ⓘ
precedes	matrix mechanics ⓘ wave mechanics ⓘ
quantizes	action integrals over closed paths ⓘ
refines	Bohr model ⓘ linked to: Bohr model of the atom
relatedTo	EBK quantization ⓘ Sommerfeld fine-structure formula ⓘ WKB approximation ⓘ
status	historically important but superseded ⓘ
typeOf	semi-classical quantization ⓘ
usesConcept	action variable ⓘ adiabatic invariant ⓘ azimuthal quantum number ⓘ principal quantum number ⓘ radial quantum number ⓘ
usesMathematicalForm	∮ p_i dq_i = n_i h ⓘ
yearProposed	1916 ⓘ

How these facts were elicited