Physics:Quantum mechanics - Revision history

WikiHarold: Fix remaining Quantum reference errors

2026-05-24T00:42:50Z

Fix remaining Quantum reference errors

← Older revision		Revision as of 00:42, 24 May 2026
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	== Explanatory notes ==		== Explanatory notes ==

			= Notes =
			{{reflist\|group=note}}

	= References =		= References =

WikiHarold: Restore missing Quantum reference definitions

2026-05-24T00:31:41Z

Restore missing Quantum reference definitions

← Older revision		Revision as of 00:31, 24 May 2026
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	One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its momentum.<ref name="Zwiebach2022" />{{rp\|427–435}}		One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its momentum.<ref name="Zwiebach2022" />{{rp\|427–435}}
	[[File:Double-slit.svg\|thumb\|left\|upright=1.2\|An illustration of the double-slit experiment]]		[[File:Double-slit.svg\|thumb\|left\|upright=1.2\|An illustration of the double-slit experiment]]
	Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference, which is often illustrated with the double-slit experiment. In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.<ref name="Lederman">{{cite book \|last1=Lederman \|first1=Leon M. \|url=https://books.google.com/books?id=qY_yOwHg_WYC&pg=PA102 \|title=Quantum Physics for Poets \|first2=Christopher T. \|last2=Hill \|publisher=Prometheus Books \|year=2011 \|isbn=978-1-61614-281-0 \|location=US}}</ref>{{rp\|102–111}}<ref name="Feynman" />{{rp\|1.1–1.8}} The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.<ref name="Lederman" /> However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave).<ref name="Lederman" />{{rp\|109}}<ref name="Müller-Kirsten">{{cite book \|last=Müller-Kirsten \|first=H. J. W. \|url=https://books.google.com/books?id=p1_Z81Le58MC&pg=PA14 \|title=Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral \|publisher=World Scientific \|year=2006 \|isbn=978-981-256-691-1 \|location=US \|page=14}}</ref><ref name="Plotnitsky">{{cite book \|last=Plotnitsky \|first=Arkady \|url=https://books.google.com/books?id=dmdUp97S4AYC&pg=PA75 \|title=Niels Bohr and Complementarity: An Introduction \|publisher=Springer \|year=2012 \|isbn=978-1-4614-4517-3 \|location=US \|pages=75–76}}</ref> However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as wave–particle duality. In addition to light, electrons, atoms, and molecules are all found to exhibit the same dual behavior when fired towards a double slit.<ref name="Feynman" />		Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference, which is often illustrated with the double-slit experiment. In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.<ref name="Lederman">{{cite book \|last1=Lederman \|first1=Leon M. \|url=https://books.google.com/books?id=qY_yOwHg_WYC&pg=PA102 \|title=Quantum Physics for Poets \|first2=Christopher T. \|last2=Hill \|publisher=Prometheus Books \|year=2011 \|isbn=978-1-61614-281-0 \|location=US}}</ref>{{rp\|102–111}}<ref name="Feynman">{{cite book
			\| last1 = Feynman
			\| first1 = Richard
			\| last2 = Leighton
			\| first2 = Robert
			\| last3 = Sands
			\| first3 = Matthew
			\| title = The Feynman Lectures on Physics
			\| volume = 3
			\| publisher = California Institute of Technology
			\| date = 1964
			\| url = https://feynmanlectures.caltech.edu/III_01.html
			\| isbn = 978-0201500646
			\| access-date = 19 December 2020
			}}</ref>{{rp\|1.1–1.8}} The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.<ref name="Lederman" /> However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave).<ref name="Lederman" />{{rp\|109}}<ref name="Müller-Kirsten">{{cite book \|last=Müller-Kirsten \|first=H. J. W. \|url=https://books.google.com/books?id=p1_Z81Le58MC&pg=PA14 \|title=Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral \|publisher=World Scientific \|year=2006 \|isbn=978-981-256-691-1 \|location=US \|page=14}}</ref><ref name="Plotnitsky">{{cite book \|last=Plotnitsky \|first=Arkady \|url=https://books.google.com/books?id=dmdUp97S4AYC&pg=PA75 \|title=Niels Bohr and Complementarity: An Introduction \|publisher=Springer \|year=2012 \|isbn=978-1-4614-4517-3 \|location=US \|pages=75–76}}</ref> However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as wave–particle duality. In addition to light, electrons, atoms, and molecules are all found to exhibit the same dual behavior when fired towards a double slit.<ref name="Feynman" />
	[[File:Quantum_book1_mechanics_yellow.png\|left\|thumb\|upright=1.2\|A simplified diagram of quantum tunneling, a phenomenon by which a particle may move through a barrier which would be impossible under classical mechanics]]		[[File:Quantum_book1_mechanics_yellow.png\|left\|thumb\|upright=1.2\|A simplified diagram of quantum tunneling, a phenomenon by which a particle may move through a barrier which would be impossible under classical mechanics]]
	Another non-classical phenomenon predicted by quantum mechanics is [[Physics:Quantum tunnelling\|quantum tunnelling]]: a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential.<ref>{{cite book \|first=David J. \|last=Griffiths \|title=Introduction to Quantum Mechanics \|title-link=Introduction to Quantum Mechanics (book) \|date=1995 \|publisher=Prentice Hall \|isbn=0-13-124405-1}}</ref> In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay, nuclear fusion in stars, and applications such as scanning tunnelling microscopy, tunnel diode and tunnel field-effect transistor.<ref name="Trixler2013">{{cite journal \|last=Trixler \|first=F. \|title=Quantum tunnelling to the origin and evolution of life \|journal=Current Organic Chemistry \|date=2013 \|volume=17 \|number=16 \|pages=1758–1770 \|doi=10.2174/13852728113179990083 \|pmid=24039543 \|pmc=3768233}}</ref><ref>{{Cite news \|last=Phifer \|first=Arnold \|date=2012-03-27 \|title=Developing more energy-efficient transistors through quantum tunneling \|url=https://news.nd.edu/news/developing-more-energy-efficient-transistors-through-quantum-tunneling/ \|access-date=2024-06-07 \|work=Notre Dame News}}</ref>		Another non-classical phenomenon predicted by quantum mechanics is [[Physics:Quantum tunnelling\|quantum tunnelling]]: a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential.<ref>{{cite book \|first=David J. \|last=Griffiths \|title=Introduction to Quantum Mechanics \|title-link=Introduction to Quantum Mechanics (book) \|date=1995 \|publisher=Prentice Hall \|isbn=0-13-124405-1}}</ref> In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay, nuclear fusion in stars, and applications such as scanning tunnelling microscopy, tunnel diode and tunnel field-effect transistor.<ref name="Trixler2013">{{cite journal \|last=Trixler \|first=F. \|title=Quantum tunnelling to the origin and evolution of life \|journal=Current Organic Chemistry \|date=2013 \|volume=17 \|number=16 \|pages=1758–1770 \|doi=10.2174/13852728113179990083 \|pmid=24039543 \|pmc=3768233}}</ref><ref>{{Cite news \|last=Phifer \|first=Arnold \|date=2012-03-27 \|title=Developing more energy-efficient transistors through quantum tunneling \|url=https://news.nd.edu/news/developing-more-energy-efficient-transistors-through-quantum-tunneling/ \|access-date=2024-06-07 \|work=Notre Dame News}}</ref>
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	== Explanatory notes ==		== Explanatory notes ==

	== References ==		= References =
	{{reflist}}		{{reflist}}

WikiHarold: Remove imported red links from Quantum page

2026-05-23T23:47:42Z

Remove imported red links from Quantum page

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	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
	{{Quantum article nav\|previous=Physics:Quantum Observables and operators\|previous label=Observables and operators\|next=Physics:Quantum mechanics measurements\|next label=Mechanics measurements}}		{{Quantum article nav\|previous=Physics:Quantum Observables and operators\|previous label=Observables and operators\|next=Physics:Quantum mechanics measurements\|next label=Mechanics measurements}}
	~~{{Quantum mechanics}}~~
	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

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	=== Harmonic oscillator ===		=== Harmonic oscillator ===
	~~{{Main\|Physics:Quantum harmonic oscillator}}~~

	[[File:QuantumHarmonicOscillatorAnimation.gif\|thumb\|upright=1.35\|right\|Some trajectories of a [[harmonic oscillator]] (i.e. a ball attached to a spring) in classical mechanics (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a wave (called the wave function), with the real part shown in blue and the imaginary part shown in red. Some of the trajectories (such as C, D, E, and F) are standing waves (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have ''any'' energy.]]		[[File:QuantumHarmonicOscillatorAnimation.gif\|thumb\|upright=1.35\|right\|Some trajectories of a [[harmonic oscillator]] (i.e. a ball attached to a spring) in classical mechanics (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a wave (called the wave function), with the real part shown in blue and the imaginary part shown in red. Some of the trajectories (such as C, D, E, and F) are standing waves (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have ''any'' energy.]]

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	* {{cite SEP \|url-id=qt-issues \|title=Philosophical Issues in Quantum Theory}}		* {{cite SEP \|url-id=qt-issues \|title=Philosophical Issues in Quantum Theory}}
	{{Physics-footer}}		{{Physics-footer}}

	~~[[Category:Quantum mechanics\| ]]~~

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

	{{Sourceattribution\|Physics:Quantum mechanics\|1}}		{{Sourceattribution\|Physics:Quantum mechanics\|1}}

WikiHarold: Clean Quantum page image and red links

2026-05-23T23:34:50Z

Clean Quantum page image and red links

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	* {{cite SEP \|url-id=qm \|title=Quantum Mechanics \|last=Ismael \|first=Jenann}}		* {{cite SEP \|url-id=qm \|title=Quantum Mechanics \|last=Ismael \|first=Jenann}}
	* {{cite SEP \|url-id=qt-issues \|title=Philosophical Issues in Quantum Theory}}		* {{cite SEP \|url-id=qt-issues \|title=Philosophical Issues in Quantum Theory}}

	~~{{Quantum mechanics topics}}~~
	{{Physics-footer}}		{{Physics-footer}}

WikiHarold: Use Quantum See also index module

2026-05-23T22:22:17Z

Use Quantum See also index module

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	== See also ==		== See also ==
	{{~~cols~~\|~~colwidth=35em}}~~		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}
	* Bra–ket notation
	* Einstein's thought experiments
	* List of textbooks on classical mechanics and quantum mechanics
	* Macroscopic quantum phenomena
	* Phase-space formulation
	* Regularization
	* Two-state quantum system
	~~{{colend~~}}

	== Explanatory notes ==		== Explanatory notes ==

	~~== See also ==~~
	~~{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}~~

	~~{{reflist\|group=note}}~~

	== References ==		== References ==

Maintenance script: Point hydrogen atom link to Quantum Collection page

2026-05-23T10:40:46Z

Point hydrogen atom link to Quantum Collection page

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	Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian.<ref name="Zwiebach2022">{{cite book \|first=Barton \|last=Zwiebach \|title=Mastering Quantum Mechanics: Essentials, Theory, and Applications \|publisher=MIT Press \|year=2022 \|isbn=978-0-262-04613-8}}</ref>{{rp\|133–137}} Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an ''s'' orbital (Fig. 1).		Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian.<ref name="Zwiebach2022">{{cite book \|first=Barton \|last=Zwiebach \|title=Mastering Quantum Mechanics: Essentials, Theory, and Applications \|publisher=MIT Press \|year=2022 \|isbn=978-0-262-04613-8}}</ref>{{rp\|133–137}} Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an ''s'' orbital (Fig. 1).

	Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]], the particle in a box, the dihydrogen cation, and the [[Physics:~~Hydrogen atom~~\|hydrogen atom]]. Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form.<ref>{{Cite journal \|last1=Zhang \|first1=Ruiqin \|last2=Deng \|first2=Conghao \|date=1993 \|title=Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems \|journal=Physical Review A \|volume=47 \|issue=1 \|pages=71–77 \|doi=10.1103/PhysRevA.47.71 \|pmid=9908895 \|bibcode=1993PhRvA..47...71Z \|issn=1050-2947}}</ref><ref>{{Cite journal \|last1=Li \|first1=Jing \|last2=Drummond \|first2=N. D. \|last3=Schuck \|first3=Peter \|last4=Olevano \|first4=Valerio \|date=2019-04-01 \|title=Comparing many-body approaches against the helium atom exact solution \|journal=SciPost Physics \|volume=6 \|issue=4 \|page=40 \|doi=10.21468/SciPostPhys.6.4.040 \|doi-access=free \|arxiv=1801.09977 \|bibcode=2019ScPP....6...40L \|issn=2542-4653}}</ref><ref>{{cite book \|last=Drake \|first=Gordon W. F. \|chapter=High Precision Calculations for Helium \|date=2023 \|title=Springer Handbook of Atomic, Molecular, and Optical Physics \|series=Springer Handbooks \|pages=199–216 \|editor-last=Drake \|editor-first=Gordon W. F. \|place=Cham \|publisher=Springer International Publishing \|doi=10.1007/978-3-030-73893-8_12 \|isbn=978-3-030-73892-1}}</ref>		Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]], the particle in a box, the dihydrogen cation, and the [[Physics:Quantum atoms/hydrogen\|hydrogen atom]]. Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form.<ref>{{Cite journal \|last1=Zhang \|first1=Ruiqin \|last2=Deng \|first2=Conghao \|date=1993 \|title=Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems \|journal=Physical Review A \|volume=47 \|issue=1 \|pages=71–77 \|doi=10.1103/PhysRevA.47.71 \|pmid=9908895 \|bibcode=1993PhRvA..47...71Z \|issn=1050-2947}}</ref><ref>{{Cite journal \|last1=Li \|first1=Jing \|last2=Drummond \|first2=N. D. \|last3=Schuck \|first3=Peter \|last4=Olevano \|first4=Valerio \|date=2019-04-01 \|title=Comparing many-body approaches against the helium atom exact solution \|journal=SciPost Physics \|volume=6 \|issue=4 \|page=40 \|doi=10.21468/SciPostPhys.6.4.040 \|doi-access=free \|arxiv=1801.09977 \|bibcode=2019ScPP....6...40L \|issn=2542-4653}}</ref><ref>{{cite book \|last=Drake \|first=Gordon W. F. \|chapter=High Precision Calculations for Helium \|date=2023 \|title=Springer Handbook of Atomic, Molecular, and Optical Physics \|series=Springer Handbooks \|pages=199–216 \|editor-last=Drake \|editor-first=Gordon W. F. \|place=Cham \|publisher=Springer International Publishing \|doi=10.1007/978-3-030-73893-8_12 \|isbn=978-3-030-73892-1}}</ref>

	However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy.<ref name="Zwiebach2022" />{{rp\|793}} Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.<ref name="Zwiebach2022" />{{rp\|849}}		However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy.<ref name="Zwiebach2022" />{{rp\|793}} Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.<ref name="Zwiebach2022" />{{rp\|849}}
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	Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, [[Physics:Quantum electrodynamics\|quantum electrodynamics]], provides a fully quantum description of the electromagnetic interaction. Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised.<ref>{{cite book \|url=https://books.google.com/books?id=6a-agBFWuyQC&pg=PA61 \|title=The Nature of Space and Time \|date=2010 \|isbn=978-1-4008-3474-7 \|last1=Hawking \|first1=Stephen \|last2=Penrose \|first2=Roger \|publisher=Princeton University Press}}</ref><ref>{{cite journal \|last1=Aoyama \|first1=Tatsumi \|last2=Hayakawa \|first2=Masashi \|last3=Kinoshita \|first3=Toichiro \|last4=Nio \|first4=Makiko \|year=2012 \|title=Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant \|journal=Physical Review Letters \|volume=109 \|issue=11 \|article-number=111807 \|arxiv=1205.5368 \|bibcode=2012PhRvL.109k1807A \|doi=10.1103/PhysRevLett.109.111807 \|pmid=23005618 \|s2cid=14712017}}</ref>		Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, [[Physics:Quantum electrodynamics\|quantum electrodynamics]], provides a fully quantum description of the electromagnetic interaction. Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised.<ref>{{cite book \|url=https://books.google.com/books?id=6a-agBFWuyQC&pg=PA61 \|title=The Nature of Space and Time \|date=2010 \|isbn=978-1-4008-3474-7 \|last1=Hawking \|first1=Stephen \|last2=Penrose \|first2=Roger \|publisher=Princeton University Press}}</ref><ref>{{cite journal \|last1=Aoyama \|first1=Tatsumi \|last2=Hayakawa \|first2=Masashi \|last3=Kinoshita \|first3=Toichiro \|last4=Nio \|first4=Makiko \|year=2012 \|title=Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant \|journal=Physical Review Letters \|volume=109 \|issue=11 \|article-number=111807 \|arxiv=1205.5368 \|bibcode=2012PhRvL.109k1807A \|doi=10.1103/PhysRevLett.109.111807 \|pmid=23005618 \|s2cid=14712017}}</ref>

	The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the [[Physics:~~Hydrogen atom~~\|hydrogen atom]] describes the electric field of the hydrogen atom using a classical <math>\textstyle -e^2/(4 \pi\epsilon_{_0}r)</math> Coulomb potential.<ref name="Zwiebach2022" />{{rp\|285}} Likewise, in a Stern–Gerlach experiment, a charged particle is modeled as a quantum system, while the background magnetic field is described classically.<ref name="Peres1993" />{{rp\|26}} This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.		The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the [[Physics:Quantum atoms/hydrogen\|hydrogen atom]] describes the electric field of the hydrogen atom using a classical <math>\textstyle -e^2/(4 \pi\epsilon_{_0}r)</math> Coulomb potential.<ref name="Zwiebach2022" />{{rp\|285}} Likewise, in a Stern–Gerlach experiment, a charged particle is modeled as a quantum system, while the background magnetic field is described classically.<ref name="Peres1993" />{{rp\|26}} This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

	Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called [[Physics:Quantum chromodynamics\|quantum chromodynamics]], and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg.<ref>{{cite web \|url=http://nobelprize.org/nobel_prizes/physics/laureates/1979/index.html \|title=The Nobel Prize in Physics 1979 \|publisher=Nobel Foundation \|access-date=16 December 2020}}</ref>		Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called [[Physics:Quantum chromodynamics\|quantum chromodynamics]], and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg.<ref>{{cite web \|url=http://nobelprize.org/nobel_prizes/physics/laureates/1979/index.html \|title=The Nobel Prize in Physics 1979 \|publisher=Nobel Foundation \|access-date=16 December 2020}}</ref>

Maintenance script: Normalize quantum page header order

2026-05-22T11:30:57Z

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	{{Quantum article nav\|previous=Physics:Quantum Observables and operators\|previous label=Observables and operators\|next=Physics:Quantum mechanics measurements\|next label=Mechanics measurements}}		{{Quantum article nav\|previous=Physics:Quantum Observables and operators\|previous label=Observables and operators\|next=Physics:Quantum mechanics measurements\|next label=Mechanics measurements}}
	{{Quantum mechanics}}		{{Quantum mechanics}}

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Maintenance script: Clean Book label remnants and backlink spacing

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Clean Book label remnants and backlink spacing

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	{{Short description\|Description of physical properties at the atomic and subatomic scale}}		{{Short description\|Description of physical properties at the atomic and subatomic scale}}

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
	{{Quantum article nav\|previous=Physics:Quantum Observables and operators\|previous label=Observables and operators\|next=Physics:Quantum mechanics measurements\|next label=Mechanics measurements}}		{{Quantum article nav\|previous=Physics:Quantum Observables and operators\|previous label=Observables and operators\|next=Physics:Quantum mechanics measurements\|next label=Mechanics measurements}}

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	{{Short description\|Description of physical properties at the atomic and subatomic scale}}~~Book I~~		{{Short description\|Description of physical properties at the atomic and subatomic scale}}

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}

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Clean Short description prefix and add Book I label

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	~~~~{{Short description\|Description of physical properties at the atomic and subatomic scale}}		{{Short description\|Description of physical properties at the atomic and subatomic scale}}Book I

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}

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	Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian.<ref name="Zwiebach2022">{{cite book \|first=Barton \|last=Zwiebach \|title=Mastering Quantum Mechanics: Essentials, Theory, and Applications \|publisher=MIT Press \|year=2022 \|isbn=978-0-262-04613-8}}</ref>{{rp\|133–137}} Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an ''s'' orbital (Fig. 1).		Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian.<ref name="Zwiebach2022">{{cite book \|first=Barton \|last=Zwiebach \|title=Mastering Quantum Mechanics: Essentials, Theory, and Applications \|publisher=MIT Press \|year=2022 \|isbn=978-0-262-04613-8}}</ref>{{rp\|133–137}} Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an ''s'' orbital (Fig. 1).

	Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]], the particle in a box, the dihydrogen cation, and the [[Physics:~~Hydrogen atom~~\|hydrogen atom]]. Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form.<ref>{{Cite journal \|last1=Zhang \|first1=Ruiqin \|last2=Deng \|first2=Conghao \|date=1993 \|title=Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems \|journal=Physical Review A \|volume=47 \|issue=1 \|pages=71–77 \|doi=10.1103/PhysRevA.47.71 \|pmid=9908895 \|bibcode=1993PhRvA..47...71Z \|issn=1050-2947}}</ref><ref>{{Cite journal \|last1=Li \|first1=Jing \|last2=Drummond \|first2=N. D. \|last3=Schuck \|first3=Peter \|last4=Olevano \|first4=Valerio \|date=2019-04-01 \|title=Comparing many-body approaches against the helium atom exact solution \|journal=SciPost Physics \|volume=6 \|issue=4 \|page=40 \|doi=10.21468/SciPostPhys.6.4.040 \|doi-access=free \|arxiv=1801.09977 \|bibcode=2019ScPP....6...40L \|issn=2542-4653}}</ref><ref>{{cite book \|last=Drake \|first=Gordon W. F. \|chapter=High Precision Calculations for Helium \|date=2023 \|title=Springer Handbook of Atomic, Molecular, and Optical Physics \|series=Springer Handbooks \|pages=199–216 \|editor-last=Drake \|editor-first=Gordon W. F. \|place=Cham \|publisher=Springer International Publishing \|doi=10.1007/978-3-030-73893-8_12 \|isbn=978-3-030-73892-1}}</ref>		Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]], the particle in a box, the dihydrogen cation, and the [[Physics:Quantum atoms/hydrogen\|hydrogen atom]]. Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form.<ref>{{Cite journal \|last1=Zhang \|first1=Ruiqin \|last2=Deng \|first2=Conghao \|date=1993 \|title=Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems \|journal=Physical Review A \|volume=47 \|issue=1 \|pages=71–77 \|doi=10.1103/PhysRevA.47.71 \|pmid=9908895 \|bibcode=1993PhRvA..47...71Z \|issn=1050-2947}}</ref><ref>{{Cite journal \|last1=Li \|first1=Jing \|last2=Drummond \|first2=N. D. \|last3=Schuck \|first3=Peter \|last4=Olevano \|first4=Valerio \|date=2019-04-01 \|title=Comparing many-body approaches against the helium atom exact solution \|journal=SciPost Physics \|volume=6 \|issue=4 \|page=40 \|doi=10.21468/SciPostPhys.6.4.040 \|doi-access=free \|arxiv=1801.09977 \|bibcode=2019ScPP....6...40L \|issn=2542-4653}}</ref><ref>{{cite book \|last=Drake \|first=Gordon W. F. \|chapter=High Precision Calculations for Helium \|date=2023 \|title=Springer Handbook of Atomic, Molecular, and Optical Physics \|series=Springer Handbooks \|pages=199–216 \|editor-last=Drake \|editor-first=Gordon W. F. \|place=Cham \|publisher=Springer International Publishing \|doi=10.1007/978-3-030-73893-8_12 \|isbn=978-3-030-73892-1}}</ref>

	However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy.<ref name="Zwiebach2022" />{{rp\|793}} Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.<ref name="Zwiebach2022" />{{rp\|849}}		However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy.<ref name="Zwiebach2022" />{{rp\|793}} Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.<ref name="Zwiebach2022" />{{rp\|849}}
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	Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, [[Physics:Quantum electrodynamics\|quantum electrodynamics]], provides a fully quantum description of the electromagnetic interaction. Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised.<ref>{{cite book \|url=https://books.google.com/books?id=6a-agBFWuyQC&pg=PA61 \|title=The Nature of Space and Time \|date=2010 \|isbn=978-1-4008-3474-7 \|last1=Hawking \|first1=Stephen \|last2=Penrose \|first2=Roger \|publisher=Princeton University Press}}</ref><ref>{{cite journal \|last1=Aoyama \|first1=Tatsumi \|last2=Hayakawa \|first2=Masashi \|last3=Kinoshita \|first3=Toichiro \|last4=Nio \|first4=Makiko \|year=2012 \|title=Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant \|journal=Physical Review Letters \|volume=109 \|issue=11 \|article-number=111807 \|arxiv=1205.5368 \|bibcode=2012PhRvL.109k1807A \|doi=10.1103/PhysRevLett.109.111807 \|pmid=23005618 \|s2cid=14712017}}</ref>		Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, [[Physics:Quantum electrodynamics\|quantum electrodynamics]], provides a fully quantum description of the electromagnetic interaction. Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised.<ref>{{cite book \|url=https://books.google.com/books?id=6a-agBFWuyQC&pg=PA61 \|title=The Nature of Space and Time \|date=2010 \|isbn=978-1-4008-3474-7 \|last1=Hawking \|first1=Stephen \|last2=Penrose \|first2=Roger \|publisher=Princeton University Press}}</ref><ref>{{cite journal \|last1=Aoyama \|first1=Tatsumi \|last2=Hayakawa \|first2=Masashi \|last3=Kinoshita \|first3=Toichiro \|last4=Nio \|first4=Makiko \|year=2012 \|title=Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant \|journal=Physical Review Letters \|volume=109 \|issue=11 \|article-number=111807 \|arxiv=1205.5368 \|bibcode=2012PhRvL.109k1807A \|doi=10.1103/PhysRevLett.109.111807 \|pmid=23005618 \|s2cid=14712017}}</ref>

	The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the [[Physics:~~Hydrogen atom~~\|hydrogen atom]] describes the electric field of the hydrogen atom using a classical <math>\textstyle -e^2/(4 \pi\epsilon_{_0}r)</math> Coulomb potential.<ref name="Zwiebach2022" />{{rp\|285}} Likewise, in a Stern–Gerlach experiment, a charged particle is modeled as a quantum system, while the background magnetic field is described classically.<ref name="Peres1993" />{{rp\|26}} This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.		The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the [[Physics:Quantum atoms/hydrogen\|hydrogen atom]] describes the electric field of the hydrogen atom using a classical <math>\textstyle -e^2/(4 \pi\epsilon_{_0}r)</math> Coulomb potential.<ref name="Zwiebach2022" />{{rp\|285}} Likewise, in a Stern–Gerlach experiment, a charged particle is modeled as a quantum system, while the background magnetic field is described classically.<ref name="Peres1993" />{{rp\|26}} This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

	Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called [[Physics:Quantum chromodynamics\|quantum chromodynamics]], and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg.<ref>{{cite web \|url=http://nobelprize.org/nobel_prizes/physics/laureates/1979/index.html \|title=The Nobel Prize in Physics 1979 \|publisher=Nobel Foundation \|access-date=16 December 2020}}</ref>		Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called [[Physics:Quantum chromodynamics\|quantum chromodynamics]], and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg.<ref>{{cite web \|url=http://nobelprize.org/nobel_prizes/physics/laureates/1979/index.html \|title=The Nobel Prize in Physics 1979 \|publisher=Nobel Foundation \|access-date=16 December 2020}}</ref>

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	{{Short description\|Description of physical properties at the atomic and subatomic scale}}		{{Short description\|Description of physical properties at the atomic and subatomic scale}}

	{{Quantum book backlink\|Foundations}}		{{Quantum book backlink\|Foundations}}
	{{Quantum article nav\|previous=Physics:Quantum Observables and operators\|previous label=Observables and operators\|next=Physics:Quantum mechanics measurements\|next label=Mechanics measurements}}		{{Quantum article nav\|previous=Physics:Quantum Observables and operators\|previous label=Observables and operators\|next=Physics:Quantum mechanics measurements\|next label=Mechanics measurements}}