Editing Quantum number (section)

== History ==
{{see also | History of quantum mechanics}}

===Electronic quantum numbers===
In the era of the [[old quantum theory]], starting from [[Max Planck]]'s proposal of quanta in his model of [[blackbody radiation]] (1900) and [[Albert Einstein]]'s adaptation of the concept to explain the [[photoelectric effect]] (1905), and until [[Erwin Schrödinger]] published his eigenfunction equation in 1926,<ref name="schrodinger">{{cite journal |author=Schrödinger, Erwin |year=1926 |title=Quantisation as an Eigenvalue Problem |journal=Annalen der Physik |volume=81 |issue=18 |pages=109–139 |bibcode=1926AnP...386..109S |doi=10.1002/andp.19263861802}}</ref> the concept behind quantum numbers developed based on atomic spectroscopy and theories from classical mechanics with extra ad hoc constraints.<ref name="Whittaker">{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York}}</ref>{{rp|106}} Many results from atomic spectroscopy had been summarized in the [[Rydberg formula]] involving differences between two series of energies related by integer steps. The [[Bohr atom|model of the atom]], first proposed by [[Niels Bohr]] in 1913, relied on a single quantum number. Together with Bohr's constraint that radiation absorption is not classical, it was able to explain the [[Balmer series]] portion of Rydberg's atomic spectrum formula.<ref>{{Cite journal |last=Heilbron |first=John L. |date=June 2013 |title=The path to the quantum atom |url=https://www.nature.com/articles/498027a |journal=Nature |language=en |volume=498 |issue=7452 |pages=27–30 |doi=10.1038/498027a |pmid=23739408 |issn=0028-0836}}</ref>

As Bohr notes in his subsequent Nobel lecture, the next step was taken by [[Arnold Sommerfeld]] in 1915.<ref>[https://www.nobelprize.org/prizes/physics/1922/bohr/lecture/ Niels Bohr – Nobel Lecture]. NobelPrize.org. Nobel Prize Outreach AB 2024. Sun. 25 Feb 2024.</ref> Sommerfeld's atomic model added a second quantum number and the concept of quantized phase integrals to justify them.<ref>{{Cite book |last1=Eckert |first1=Michael |title=Arnold Sommerfeld: science, life and turbulent times 1868-1951 |last2=Eckert |first2=Michael |last3=Artin |first3=Tom |date=2013 |publisher=Springer |isbn=978-1-4614-7461-6 |location=New York}}</ref>{{rp|207}} Sommerfeld's model was still essentially two dimensional, modeling the electron as orbiting in a plane; in 1919 he extended his work to three dimensions using 'space quantization' in place of the quantized phase integrals.<ref name=Kragh2012Bohr>{{Cite book |last=Kragh |first=Helge |url=http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199654987.001.0001/acprof-9780199654987 |title=Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925 |date=2012-05-17 |publisher=Oxford University Press |isbn=978-0-19-965498-7 |doi=10.1093/acprof:oso/9780199654987.003.0004}}</ref>{{rp|152}} [[Karl Schwarzschild]] and Sommerfeld's student, [[Paul Epstein]], independently showed that adding third quantum number gave a complete account for the [[Stark effect]] results.

A consequence of space quantization was that the electron's orbital interaction with an external magnetic field would be quantized. This seemed to be confirmed when the results of the [[Stern-Gerlach]] experiment reported quantized results for silver atoms in an inhomogeneous magnetic field. The confirmation would turn out to be premature: more quantum numbers would be needed.<ref name="FriedrichHerschbach">{{Cite journal |last1=Friedrich |first1=Bretislav |last2=Herschbach |first2=Dudley |date=2003-12-01 |title=Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics |url=https://pubs.aip.org/physicstoday/article/56/12/53/632269/Stern-and-Gerlach-How-a-Bad-Cigar-Helped-Reorient |journal=Physics Today |language=en |volume=56 |issue=12 |pages=53–59 |doi=10.1063/1.1650229 |bibcode=2003PhT....56l..53F |issn=0031-9228|url-access=subscription }}</ref>

The fourth and fifth quantum numbers of the atomic era arose from attempts to understand the [[Zeeman effect]]. Like the Stern-Gerlach experiment, the Zeeman effect reflects the interaction of atoms with a magnetic field; in a weak field the experimental results were called "anomalous", they diverged from any theory at the time. [[Wolfgang Pauli]]'s solution to this issue was to introduce another quantum number taking only two possible values, <math>\pm \hbar/2</math>.<ref name=Giulini>{{Cite journal |last=Giulini |first=Domenico |date=2008-09-01 |title=Electron spin or "classically non-describable two-valuedness" |url=https://www.sciencedirect.com/science/article/pii/S1355219808000269 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |volume=39 |issue=3 |pages=557–578 |doi=10.1016/j.shpsb.2008.03.005 |issn=1355-2198|arxiv=0710.3128 |bibcode=2008SHPMP..39..557G |hdl=11858/00-001M-0000-0013-13C8-1 }}</ref> This would ultimately become the quantized values of the projection of [[spin (physics)|spin]], an intrinsic angular momentum quantum of the electron. In 1927 Ronald Fraser demonstrated that the quantization in the Stern-Gerlach experiment was due to the magnetic moment associated with the electron spin rather than its orbital angular momentum.<ref name=FriedrichHerschbach/> Pauli's success in developing the arguments for a spin quantum number without relying on classical models set the stage for the development of quantum numbers for elementary particles in the remainder of the 20th century.<ref name=Giulini/>

Bohr, with his [[Aufbau principle|Aufbau]] or "building up" principle, and Pauli with his [[Pauli exclusion principle|exclusion principle]] connected the atom's electronic quantum numbers in to a framework for predicting the properties of atoms.<ref>{{Cite book |last=Kragh |first=Helge |url=http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199654987.001.0001/acprof-9780199654987 |title=Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925 |date=2012-05-17 |publisher=Oxford University Press |isbn=978-0-19-965498-7 |language=en |doi=10.1093/acprof:oso/9780199654987.003.0007}}</ref> When Schrödinger published his [[Schrodinger equation|wave equation]] and calculated the energy levels of hydrogen, these two principles carried over to become the basis of atomic physics.

===Nuclear quantum numbers===
With successful models of the atom, the attention of physics turned to models of the nucleus. Beginning with Heisenberg's initial model of proton-neutron binding in 1932, [[Eugene Wigner]] introduced [[isospin]] in 1937, the first 'internal' quantum number unrelated to a symmetry in real space-time.<ref name=Brown1987>{{Cite book |last=Brown |first=L.M. |chapter-url=https://archive.org/details/festivalfestschr0000unse/page/40/mode/2up?q=isospin |title=Festi-Val: Festschrift for Val Telegdi; essays in physics in honour of his 65th birthday; [a symposium ... was held at CERN, Geneva on 6 July 1987] |date=1988 |publisher=North-Holland Physics Publ |isbn=978-0-444-87099-5 |editor-last=Winter |editor-first=Klaus |location=Amsterdam |language=en |chapter=Remarks on the history of isospin |editor-last2=Telegdi |editor-first2=Valentine L.}}</ref>{{rp|45}}

===Connection to symmetry===
As quantum mechanics developed, abstraction increased and models based on symmetry and invariance played increasing roles. Two years before his work on the quantum wave equation, Schrödinger applied the symmetry ideas originated by [[Emmy Noether]] and [[Hermann Weyl]] to the electromagnetic field.<ref name=Baggott40>{{Cite book |last=Baggott |first=J. E. |title=The quantum story: a history in 40 moments |date=2013 |publisher=Oxford Univ. Press |isbn=978-0-19-956684-6 |edition=Impression: 3 |location=Oxford}}</ref>{{rp|198}} As [[quantum electrodynamics]] developed in the 1930s and 1940s, [[group theory]] became an important tool. By 1953 [[Chen Ning Yang]] had become obsessed with the idea that group theory could be applied to connect the conserved quantum numbers of nuclear collisions to symmetries in a field theory of nucleons.<ref name=Baggott40/>{{rp|202}} With [[Robert Mills (physicist)|Robert Mills]], Yang developed a [[non-abelian gauge theory]] based on the conservation of the nuclear [[isospin]] quantum numbers.