Sommerfeld fine-structure formula
E234759
The Sommerfeld fine-structure formula is a relativistic extension of the Bohr model that accurately predicts the fine-structure energy levels of the hydrogen atom.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Sommerfeld fine-structure formula canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2093717 — 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 fine-structure formula Context triple: [Arnold Sommerfeld, knownFor, Sommerfeld fine-structure formula]
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A.
Rydberg constant
The Rydberg constant is a fundamental physical constant that characterizes the limiting value of the highest wavenumber (or lowest wavelength) of any photon that can be emitted from the hydrogen atom, playing a key role in atomic spectroscopy and quantum theory.
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B.
Stark effect
The Stark effect is the splitting and shifting of atomic or molecular spectral lines caused by an external electric field.
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C.
Klein–Nishina formula
The Klein–Nishina formula is a fundamental result in quantum electrodynamics that gives the differential cross section for Compton scattering of photons by free electrons, incorporating relativistic and quantum effects.
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D.
Zeeman effect
The Zeeman effect is the splitting of atomic or molecular spectral lines into multiple components when subjected to an external magnetic field, revealing information about energy levels and magnetic moments.
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E.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
- 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 fine-structure formula Target entity description: The Sommerfeld fine-structure formula is a relativistic extension of the Bohr model that accurately predicts the fine-structure energy levels of the hydrogen atom.
-
A.
Rydberg constant
The Rydberg constant is a fundamental physical constant that characterizes the limiting value of the highest wavenumber (or lowest wavelength) of any photon that can be emitted from the hydrogen atom, playing a key role in atomic spectroscopy and quantum theory.
-
B.
Stark effect
The Stark effect is the splitting and shifting of atomic or molecular spectral lines caused by an external electric field.
-
C.
Klein–Nishina formula
The Klein–Nishina formula is a fundamental result in quantum electrodynamics that gives the differential cross section for Compton scattering of photons by free electrons, incorporating relativistic and quantum effects.
-
D.
Zeeman effect
The Zeeman effect is the splitting of atomic or molecular spectral lines into multiple components when subjected to an external magnetic field, revealing information about energy levels and magnetic moments.
-
E.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
hydrogen atom energy-level formula
ⓘ
physical formula ⓘ quantum theory result ⓘ relativistic correction ⓘ |
| accuracy | accurately reproduces hydrogen fine-structure energies to order α^4 ⓘ |
| appliesTo |
hydrogen atom
ⓘ
hydrogen-like atoms ⓘ |
| approximationTo | Dirac equation energy spectrum for hydrogen ⓘ |
| assumes |
electron moving in a Coulomb field of a point nucleus
ⓘ
relativistic mechanics for electron motion ⓘ |
| context |
fine structure of hydrogen spectral lines
ⓘ
relativistic corrections in atomic spectra ⓘ |
| describes | fine structure of hydrogen energy levels ⓘ |
| developedBy | Arnold Sommerfeld ⓘ |
| energyExpression | E_{n,j} = m c^2 \left[1 + \frac{(Z\alpha)^2}{\left(n - j - 1/2 + \sqrt{(j+1/2)^2 - (Z\alpha)^2}\right)^2} \right]^{-1/2} - m c^2 ⓘ |
| era | early 20th century atomic theory ⓘ |
| extends |
Bohr model of the atom
ⓘ
surface form:
Bohr model
|
| field |
atomic physics
ⓘ
quantum mechanics ⓘ relativistic quantum theory ⓘ |
| gives |
energy levels including Darwin-term-like corrections in an effective way
ⓘ
energy levels including relativistic kinetic energy corrections ⓘ energy levels including spin–orbit coupling effects in an effective way ⓘ |
| historicalRole |
improved agreement of Bohr model with hydrogen spectra
ⓘ
precursor to Dirac theory of the electron ⓘ |
| introduces | relativistic corrections to Bohr energy levels ⓘ |
| involves |
Planck constant
ⓘ
electron rest mass ⓘ fine-structure constant ⓘ orbital angular momentum quantum number l ⓘ principal quantum number n ⓘ speed of light ⓘ total angular momentum quantum number j ⓘ |
| is | relativistic extension of the Bohr model ⓘ |
| namedAfter | Arnold Sommerfeld ⓘ |
| neglects |
finite nuclear size effects
ⓘ
quantum electrodynamics radiative corrections ⓘ |
| predicts |
dependence of energy on orbital quantum number l
ⓘ
dependence of energy on principal quantum number n ⓘ dependence of energy on total angular momentum quantum number j ⓘ fine-structure splitting of hydrogen spectral lines ⓘ |
| relatedTo |
Sommerfeld quantization rules
ⓘ
surface form:
Bohr–Sommerfeld quantization
old quantum theory ⓘ |
| uses |
Coulomb potential
ⓘ
elliptical electron orbits ⓘ special relativity ⓘ |
| validFor | low nuclear charge Z where Zα ≪ 1 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Sommerfeld fine-structure formula Description of subject: The Sommerfeld fine-structure formula is a relativistic extension of the Bohr model that accurately predicts the fine-structure energy levels of the hydrogen atom.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.