Editing Quantum mechanics (section)

== Overview and fundamental concepts ==

Quantum mechanics allows the calculation of properties and behaviour of [[physical systems]]. It is typically applied to microscopic systems: [[molecules]], [[atoms]] and [[subatomic particle]]s. It has been demonstrated to hold for complex molecules with thousands of atoms,<ref>{{cite journal |author=Fein |first1=Yaakov Y. |last2=Geyer |first2=Philipp |last3=Zwick |first3=Patrick |last4=Kiałka |first4=Filip |last5=Pedalino |first5=Sebastian |last6=Mayor |first6=Marcel |last7=Gerlich |first7=Stefan |last8=Arndt |first8=Markus |date=September 2019 |title=Quantum superposition of molecules beyond 25 kDa |journal=Nature Physics |volume=15 |issue=12 |pages=1242–1245 |bibcode=2019NatPh..15.1242F |doi=10.1038/s41567-019-0663-9 |s2cid=203638258}}</ref> but its application to human beings raises philosophical problems, such as [[Wigner's friend]], and its application to the universe as a whole remains speculative.<ref>{{cite journal |last1=Bojowald |first1=Martin |title=Quantum cosmology: a review |journal=Reports on Progress in Physics |date=2015 |volume=78 |issue=2 |page=023901 |doi=10.1088/0034-4885/78/2/023901 |pmid=25582917 |arxiv=1501.04899 |bibcode=2015RPPh...78b3901B |s2cid=18463042}}</ref> Predictions of quantum mechanics have been verified experimentally to an extremely high degree of [[accuracy]]. For example, the refinement of quantum mechanics for the interaction of light and matter, known as [[quantum electrodynamics]] (QED), has been [[Precision tests of QED|shown to agree with experiment]] to within 1 part in 10<sup>12</sup> when predicting the magnetic properties of an electron.<ref>{{cite journal |first1=X. |last1=Fan |first2=T. G. |last2=Myers |first3=B. A. D. |last3=Sukra |first4=G. |last4=Gabrielse |title=Measurement of the Electron Magnetic Moment |journal=Physical Review Letters |volume=130 |pages=071801 |date=2023-02-13 |issue=7 |doi=10.1103/PhysRevLett.130.071801 |pmid=36867820 |arxiv=2209.13084 |bibcode=2023PhRvL.130g1801F}}</ref>

A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a [[complex number]], known as a probability amplitude. This is known as the [[Born rule]], named after physicist [[Max Born]]. For example, a quantum particle like an [[electron]] can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a [[probability density function]] for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The [[Schrödinger equation]] relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.<ref name="Zwiebach2022" />{{rp|67–87}}

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]]]]
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 [[Coherence (physics)|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 [[Interference (wave propagation)|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, [[Double-slit experiment#Which way|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:QuantumTunnel.jpg|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 [[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 |author-link=David J. 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>

When quantum systems interact, the result can be the creation of [[quantum entanglement]]: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "...<em>the</em> characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought".<ref>{{cite encyclopedia |url=https://plato.stanford.edu/entries/qt-entangle/ |first=Jeffrey |last=Bub |author-link=Jeffrey Bub |title=Quantum entanglement |encyclopedia=Stanford Encyclopedia of Philosophy |title-link=Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |editor-first=Edward N. |editor-last=Zalta |year=2019}}</ref> Quantum entanglement enables [[quantum computing]] and is part of quantum communication protocols, such as [[quantum key distribution]] and [[superdense coding]].<ref name="Caves">{{cite book |first=Carlton M. |last=Caves |author-link=Carlton M. Caves |chapter=Quantum Information Science: Emerging No More |title=OSA Century of Optics |publisher=[[The Optical Society]] |arxiv=1302.1864 |bibcode=2013arXiv1302.1864C |year=2015 |isbn=978-1-943580-04-0 |pages=320–323 |editor-first1=Paul |editor-last1=Kelley |editor-first2=Govind |editor-last2=Agrawal |editor-first3=Mike |editor-last3=Bass |editor-first4=Jeff |editor-last4=Hecht |editor-first5=Carlos |editor-last5=Stroud}}</ref> Contrary to popular misconception, entanglement does not allow sending signals [[faster than light]], as demonstrated by the [[no-communication theorem]].<ref name="Caves" />

Another possibility opened by entanglement is testing for "[[hidden variable theory|hidden variables]]", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly [[Bell's theorem]], have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of <em>local</em> hidden variables, then the results of a [[Bell test]] will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables.<ref name="wiseman15">{{Cite journal |last=Wiseman |first=Howard |author-link=Howard M. Wiseman |date=October 2015 |title=Death by experiment for local realism |journal=[[Nature (journal)|Nature]] |volume=526 |issue=7575 |pages=649–650 |doi=10.1038/nature15631 |pmid=26503054 |issn=0028-0836 |doi-access=free}}</ref><ref name="wolchover17">{{Cite magazine |url=https://www.quantamagazine.org/20170207-bell-test-quantum-loophole/ |title=Experiment Reaffirms Quantum Weirdness |last=Wolchover |first=Natalie |author-link=Natalie Wolchover |date=7 February 2017 |magazine=[[Quanta Magazine]] |access-date=8 February 2020}}</ref>

It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also [[linear algebra]], [[differential equation]]s, [[group theory]], and other more advanced subjects.<ref>{{cite web |url=https://math.ucr.edu/home/baez/books.html |title=How to Learn Math and Physics |date=20 March 2020 |website=University of California, Riverside |access-date=19 December 2020 |first=John C. |last=Baez |author-link=John C. Baez |quote=there's no way to understand the interpretation of quantum mechanics without also being able to <em>solve quantum mechanics problems</em>&nbsp;– to understand the theory, you need to be able to use it (and vice versa)}}</ref><ref>{{cite book |first=Carl |last=Sagan |author-link=Carl Sagan |title=The Demon-Haunted World: Science as a Candle in the Dark |page=249 |publisher=Ballantine Books |year=1996 |isbn=0-345-40946-9 |title-link=The Demon-Haunted World |quote="For most physics students, (the "mathematical underpinning" of quantum mechanics) might occupy them from, say, third grade to early graduate school{{snd}}roughly 15 years. ... The job of the popularizer of science, trying to get across some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion{{snd}}partly for this reason.}}</ref> Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.