Editing History of atomic theory (section)

==Modern quantum mechanical models==
{{Main|History of quantum mechanics}}
[[File:S-p-Orbitals.svg|right|thumb|The five filled atomic orbitals of a neon atom separated and arranged in order of increasing energy from left to right, with the last three orbitals being [[Degenerate energy levels|equal in energy]]. Each orbital holds up to two electrons, which most probably exist in the zones represented by the colored bubbles. Each electron is equally present in both orbital zones, shown here by color only to highlight the different wave phase.]]
In 1924, [[Louis de Broglie]] proposed that all particles—particularly subatomic particles such as electrons—have an associated wave. [[Erwin Schrödinger]], fascinated by this idea, developed an equation<ref name="schrodinger">{{cite journal|author=Schrödinger, Erwin|title=Quantisation as an Eigenvalue Problem|journal=Annalen der Physik|volume=81|issue=18|pages=109–139|year=1926|doi=10.1002/andp.19263861802|bibcode = 1926AnP...386..109S }}</ref> that describes an electron as a [[wave function]] instead of a point. This approach predicted many of the spectral phenomena that Bohr's model failed to explain, but it was difficult to visualize, and faced opposition.<ref name="Mahanti">{{cite news|author=Mahanti, Subodh|url=http://www.vigyanprasar.gov.in/scientists/ESchrodinger.htm|title=Erwin Schrödinger: The Founder of Quantum Wave Mechanics|access-date=2009-08-01|url-status=dead|archive-url=https://web.archive.org/web/20090417074535/http://www.vigyanprasar.gov.in/scientists/ESchrodinger.htm|archive-date=2009-04-17}}</ref> One of its critics, [[Max Born]], proposed instead that Schrödinger's wave function did not describe the physical extent of an electron (like a charge distribution in classical electromagnetism), but rather gave the probability that an electron would, when measured, be found at a particular point.<ref>{{cite news|author=Mahanti, Subodh|url=http://www.vigyanprasar.gov.in/scientists/MBorn.htm|title=Max Born: Founder of Lattice Dynamics|access-date=2009-08-01|url-status=dead|archive-url=https://web.archive.org/web/20090122193755/http://www.vigyanprasar.gov.in/scientists/MBorn.htm|archive-date=2009-01-22}}</ref> This reconciled the ideas of wave-like and particle-like electrons: the behavior of an electron, or of any other subatomic entity, has [[Wave–particle duality|both wave-like and particle-like aspects]], and whether one aspect or the other is observed depend upon the experiment.<ref>{{cite news|author=Greiner, Walter|url=https://books.google.com/books?id=7qCMUfwoQcAC&q=wave-particle+all-particles&pg=PA29 |title=Quantum Mechanics: An Introduction|date = 4 October 2000| publisher=Springer |isbn = 9783540674580|access-date=2010-06-14}}</ref>

A consequence of describing particles as waveforms rather than points is that it is mathematically impossible to calculate with precision both the position and momentum of a particle at a given point in time. This became known as the [[uncertainty principle]], a concept first introduced by [[Werner Heisenberg]] in 1927.{{citation needed|date=October 2024}}

Schrödinger's wave model for hydrogen replaced Bohr's model, with its neat, clearly defined circular orbits. The [[Atomic orbital|modern model of the atom]] describes the positions of electrons in an atom in terms of probabilities. An electron can potentially be found at any distance from the nucleus, but, depending on its energy level and [[angular momentum]], exists more frequently in certain regions around the nucleus than others; this pattern is referred to as its [[atomic orbital]]. The orbitals come in a variety of shapes—[[sphere]], [[dumbbell]], [[torus]], etc.—with the nucleus in the middle.<ref>{{cite news|author1=Milton Orchin |author2=Roger Macomber |author3=Allan Pinhas |author4=R. Wilson |url=http://media.wiley.com/product_data/excerpt/81/04716802/0471680281.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://media.wiley.com/product_data/excerpt/81/04716802/0471680281.pdf |archive-date=2022-10-09 |url-status=live |title=The Vocabulary and Concepts of Organic Chemistry, Second Edition|access-date=2010-06-14}}</ref> The shapes of atomic orbitals are found by solving the Schrödinger equation.<ref>{{Cite book |last=Zwiebach |first=Barton |url=https://www.worldcat.org/oclc/1306066387 |title=Mastering Quantum Mechanics Essentials, Theory, and Applications. |date=2022 |publisher=MIT Press |isbn=978-0-262-36689-2 |location=Cambridge |pages=281–305 |oclc=1306066387 |author-link=Barton Zwiebach}}</ref> Analytic solutions of the Schrödinger equation are known for [[List of quantum-mechanical systems with analytical solutions|very few relatively simple model Hamiltonians]] including the [[hydrogen atom]] and the [[dihydrogen cation|hydrogen molecular ion]].<ref>{{Cite journal |last=Grivet |first=Jean-Philippe |date=January 2002 |title=The Hydrogen Molecular Ion Revisited |url=https://pubs.acs.org/doi/abs/10.1021/ed079p127 |journal=Journal of Chemical Education |language=en |volume=79 |issue=1 |pages=127 |doi=10.1021/ed079p127 |bibcode=2002JChEd..79..127G |issn=0021-9584|url-access=subscription }}</ref> Beginning with the [[helium]] atom—which contains just two electrons—numerical methods are used to solve the Schrödinger equation.<ref>{{Cite journal |last1=Levin |first1=F. S. |last2=Shertzer |first2=J. |date=1985-12-01 |title=Finite-element solution of the Schrödinger equation for the helium ground state |url=https://link.aps.org/doi/10.1103/PhysRevA.32.3285 |journal=Physical Review A |language=en |volume=32 |issue=6 |pages=3285–3290 |doi=10.1103/PhysRevA.32.3285 |pmid=9896495 |bibcode=1985PhRvA..32.3285L |issn=0556-2791|url-access=subscription }}</ref>

Qualitatively the shape of the atomic orbitals of multi-electron atoms resemble the states of the hydrogen atom. The [[Pauli principle]] requires the distribution of these electrons within the atomic orbitals such that no more than two electrons are assigned to any one orbital; this requirement profoundly affects the atomic properties and ultimately the bonding of atoms into molecules.<ref>Karplus, Martin, and Richard Needham Porter. "Atoms and molecules; an introduction for students of physical chemistry." Atoms and molecules; an introduction for students of physical chemistry (1970).</ref>{{rp|182}}