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{{Short description|Learning project introducing the quantum concept and the fundamental principles of quantum mechanics, from quantization to quantum technologies}}

{{Quantum book backlink|Foundations}}

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'''Quantum basics''' introduces the quantum world: the physical idea that energy, matter, and other measurable properties can occur in discrete packets rather than as perfectly continuous quantities. These packets are called '''quanta''', and their study leads to [[Physics:Quantum mechanics|quantum mechanics]], quantum fields, quantum information, and modern quantum technologies.

In [[Physics:Physics|physics]], a '''quantum''' ({{plural form}}: '''quanta''') is the minimum amount of any physical entity ([[Physics:Physical property|physical property]]) involved in an [[Physics:Fundamental interaction|interaction]]. Quantum is a discrete quantity of energy proportional in magnitude to the frequency of the radiation it represents. The fundamental notion that a chemical property can be "quantized" is referred to as "the hypothesis of [[Physics:Quantization|quantization]]".<ref>Wiener, N. (1966). ''Differential Space, Quantum Systems, and Prediction''. Cambridge, Massachusetts: The Massachusetts Institute of Technology Press</ref> This means that the [[Magnitude (mathematics)|magnitude]] of the physical property can take on only discrete values consisting of [[Multiple (mathematics)|integer multiples]] of one quantum. For example, a [[Physics:Photon|photon]] is a single quantum of [[Physics:Light|light]] of a specific [[Physics:Frequency|frequency]] (or of any other form of [[Physics:Electromagnetic radiation|electromagnetic radiation]]). Similarly, the energy of an [[Physics:Electron|electron]] bound within an [[Physics:Atom|atom]] is quantized and can exist only in certain discrete values. (Atoms and matter in general are stable because electrons can exist only at discrete energy levels within an atom.) Quantization is one of the foundations of the much broader physics of [[Physics:Quantum mechanics|quantum mechanics]]. Quantization of [[Physics:Energy|energy]] and its influence on how energy and matter interact ([[Physics:Quantum electrodynamics|quantum electrodynamics]]) is part of the fundamental framework for understanding and describing nature.

The quantum world is not like anything we experience in daily life. Particles can behave like waves, measurement changes what can be known, and separated systems can become entangled. This page combines the basic definition of a quantum with a broader introduction to the principles and technologies that grew from it.
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[[File:Atom 8 yellow.jpg|thumb|280px|Artist impression of an atom]]
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==Etymology and discovery==
{{Quantum field theory|cTopic=Equations}}
The word ''quantum'' is the neuter singular of the [[Social:Latin|Latin]] interrogative adjective ''quantus'', meaning "how much". "Quanta", the neuter plural, short for "quanta of electricity" (electrons), was used in a 1902 article on the [[Physics:Photoelectric effect|photoelectric effect]] by [[Biography:Philipp Lenard|Philipp Lenard]], who credited [[Biography:Hermann von Helmholtz|Hermann von Helmholtz]] for using the word in the area of electricity. However, the word ''quantum'' in general was well known before 1900,<ref>E. Cobham Brewer 1810–1897. [http://www.bartleby.com/81/13830.html Dictionary of Phrase and Fable. 1898.] {{Web archive |url=https://web.archive.org/web/20170630232946/http://www.bartleby.com/81/13830.html |date=2017-06-30 }}</ref> e.g. ''quantum'' was used in E. A. Poe's Loss of Breath. It was often used by physicians, such as in the term ''quantum satis'', "the amount which is enough". Both Helmholtz and [[Biography:Julius von Mayer|Julius von Mayer]] were physicians as well as physicists. Helmholtz used ''quantum'' with reference to heat in his article<ref>[http://www.ub.uni-heidelberg.de/helios/fachinfo/www/math/edd/helmholtz/R-Mayer.pdf E. Helmholtz, Robert Mayer's Priorität] {{Web archive |url=https://web.archive.org/web/20150929101449/http://www.ub.uni-heidelberg.de/helios/fachinfo/www/math/edd/helmholtz/R-Mayer.pdf |date=2015-09-29 }} </ref> on Mayer's work, and the word ''quantum'' can be found in the formulation of the [[Physics:First law of thermodynamics|first law of thermodynamics]] by Mayer in his letter<ref>{{cite web |url=http://fs.math.uni-frankfurt.de/fsmath/misc/RobertMayer.html |title=Heimatseite von Robert J. Mayer |last=Herrmann |first=Armin |publisher=Weltreich der Physik, Gent-Verlag |language=de |date=1991|url-status=bot: unknown |archive-url=https://web.archive.org/web/19980209044633/http://fs.math.uni-frankfurt.de/fsmath/misc/RobertMayer.html |archive-date=1998-02-09}}</ref> dated July 24, 1841.
[[File:Max Planck (1858-1947).jpg|thumb|upright=1|German [[Physicist|Physicist]] and 1918 Nobel Prize for Physics recipient Max Planck (1858–1947)]]
In 1901, [[Biography:Max Planck|Max Planck]] used ''quanta'' to mean "quanta of matter and electricity",<ref name="Planck1901">{{cite journal |last = Planck |first = M. |year = 1901 |title = Ueber die Elementarquanta der Materie und der Elektricität |journal = [[Physics:Annalen der Physik|Annalen der Physik]] |volume = 309 |pages = 564–566 |doi = 10.1002/andp.19013090311 |bibcode = 1901AnP...309..564P |issue = 3 |language = de |url = https://zenodo.org/record/1423997 |access-date = 2019-09-16 |archive-date = 2023-06-24 |archive-url = https://web.archive.org/web/20230624230014/https://zenodo.org/record/1423997 |url-status = live }}</ref> gas, and heat.<ref>{{cite journal |last1=Planck |first1=Max |title=Ueber das thermodynamische Gleichgewicht von Gasgemengen |journal=Annalen der Physik |volume=255 |pages=358–378 |year=1883 |doi=10.1002/andp.18832550612 |bibcode=1883AnP...255..358P |issue=6 |language=de |url=https://zenodo.org/record/1423794 |access-date=2019-07-05 |archive-date=2021-01-21 |archive-url=https://web.archive.org/web/20210121222137/https://zenodo.org/record/1423794 |url-status=live }}</ref> In 1905, in response to Planck's work and the experimental work of Lenard (who explained his results by using the term ''quanta of electricity''), [[Biography:Albert Einstein|Albert Einstein]] suggested that [[Physics:Radiation|radiation]] existed in spatially localized packets which he called [[Physics:Photons|"quanta of light"]] ("''Lichtquanta''").<ref name="Einstein1905">{{cite journal |last = Einstein |first = A. |author-link = Albert Einstein |year = 1905 |title = Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt |url = http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_132-148.pdf |journal = [[Physics:Annalen der Physik|Annalen der Physik]] |volume = 17 |pages = 132–148 |doi = 10.1002/andp.19053220607 |bibcode = 1905AnP...322..132E |issue = 6 |language = de |doi-access = free |access-date = 2010-08-26 |archive-date = 2015-09-24 |archive-url = https://web.archive.org/web/20150924072915/http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_132-148.pdf |url-status = live }}. A partial [https://en.wikisource.org/?curid=59468 English translation] {{Webarchive |url=https://web.archive.org/web/20210121022128/https://en.wikisource.org/?curid=59468 |date=2021-01-21 }} is available from [[Wikisource]].</ref>

The concept of quantization of radiation was discovered in 1900 by [[Biography:Max Planck|Max Planck]], who had been trying to understand the emission of radiation from heated objects, known as [[Astronomy:Black-body radiation|black-body radiation]]. By assuming that energy can be absorbed or released only in tiny, differential, discrete packets (which he called "bundles", or "energy elements"),<ref>{{cite journal |author=Max Planck |title=Ueber das Gesetz der Energieverteilung im Normalspectrum (On the Law of Distribution of Energy in the Normal Spectrum) |url=http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html |journal=Annalen der Physik |volume= 309 |doi=10.1002/andp.19013090310 |page=553 |year=1901 |archive-url = https://web.archive.org/web/20080418002757/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html |archive-date = 2008-04-18 |bibcode = 1901AnP...309..553P |issue=3 |doi-access=free }}</ref> Planck accounted for certain objects changing color when heated.<ref>Brown, T., LeMay, H., Bursten, B. (2008). ''Chemistry: The Central Science'' Upper Saddle River, New Jersey: Pearson Education {{ISBN|0-13-600617-5}}</ref> On December 14, 1900, Planck reported his [[Physics:Planck's law|findings]] to the German Physical Society, and introduced the idea of quantization for the first time as a part of his research on black-body radiation.<ref>{{cite journal |last1=Klein |first1=Martin J. |title=Max Planck and the beginnings of the quantum theory |journal=Archive for History of Exact Sciences |volume=1 |pages=459–479 |year=1961 |doi=10.1007/BF00327765 |issue=5|s2cid=121189755 }}</ref> As a result of his experiments, Planck deduced the numerical value of ''h'', known as the [[Physics:Planck constant|Planck constant]], and reported more precise values for the unit of electrical charge and the [[Physics:Avogadro constant|Avogadro&ndash;Loschmidt number]], the number of real molecules in a [[Mole (unit)|mole]], to the German Physical Society. After his theory was validated, Planck was awarded the Nobel Prize in Physics for his discovery in 1918.

== Quantization ==
{{main article| Physics:Quantization}}
While quantization was first discovered in [[Physics:Electromagnetic radiation|electromagnetic radiation]], it describes a fundamental aspect of energy not just restricted to photons.<ref>{{Cite web |last=Parker |first=Will |date=2005-02-11 |title=Real-World Quantum Effects Demonstrated |url=http://www.scienceagogo.com/news/20050110221715data_trunc_sys.shtml |access-date=2023-08-20 |website=ScienceAGoGo |language=en-US}}</ref>
In the attempt to bring theory into agreement with experiment, Max Planck postulated that electromagnetic energy is absorbed or emitted in discrete packets, or quanta.<ref>Modern Applied Physics-Tippens third edition; McGraw-Hill.</ref><!-- This source is vague -->

== History ==
Quantum mechanics emerged in the early 20th century to resolve puzzles that classical physics could not explain, such as black-body radiation, atomic spectra, and the photoelectric effect.
* In 1900, [[Biography:Max Planck|Max Planck]] proposed that energy is emitted in discrete packets called ''quanta'' to solve the ultraviolet catastrophe.
* In 1905, [[Biography:Albert Einstein|Albert Einstein]] used the quantum idea to explain the photoelectric effect, showing that light itself behaves as particles (later called photons).
* [[Biography:Niels Bohr|Niels Bohr]] incorporated quantization into his 1913 model of the atom, explaining atomic emission lines.
* The complete mathematical framework arrived in the mid-1920s: [[Biography:Werner Heisenberg|Werner Heisenberg]] developed matrix mechanics (1925), [[Biography:Erwin Schrödinger|Erwin Schrödinger]] formulated wave mechanics (1926), and [[Biography:Max Born|Max Born]] introduced the probabilistic interpretation of the wave function.
* [[Biography:Paul Dirac|Paul Dirac]] later unified quantum mechanics with special relativity, laying the groundwork for quantum field theory.

== Foundational developments ==
Transformed physics and enabled technologies like transistors, lasers, and MRI.
In recent decades, experimental advances have allowed direct observation of ultrafast quantum processes on attosecond (10−18 s) timescales.
* [[Biography:Henry Kapteyn|'''Henry C. Kapteyn''']], together with [[Wikipedia:Margaret Murnane|'''Margaret M. Murnane''']], pioneered high-harmonic generation (HHG) techniques that produce coherent extreme-ultraviolet and soft X-ray light using tabletop lasers, generating attosecond pulses that capture electron motion in real time.<ref>{{cite web |url=https://fi.edu/en/awards/laureates/henry-c-kapteyn |title=Henry C. Kapteyn |publisher=The Franklin Institute |access-date=February 10, 2026}}</ref><ref>{{cite web |url=https://jila.colorado.edu/kmgroup |title=About the Kapteyn-Murnane Group |publisher=JILA, University of Colorado Boulder |access-date=February 10, 2026}}</ref>

* [[Biography:Andrew N. Jordan|'''Andrew N. Jordan''']] has advanced quantum measurement theory, particularly weak measurements, continuous quantum measurements, and precision limits, providing deeper insight into the measurement process itself and enabling new quantum sensing technologies.<ref>{{cite book |last1=Jordan |first1=Andrew N. |last2=Siddiqi |first2=Irfan |title=Quantum Measurement: Theory and Practice |publisher=Cambridge University Press |year=2024 |isbn=9781009100069 |url=https://www.cambridge.org/core/books/quantum-measurement/68FF73B5B4FEE63FB16441FED12D322C}}</ref><ref>{{cite web |url=https://www.chapman.edu/our-faculty/andrew-jordan.aspx |title=Dr. Andrew Jordan |publisher=Chapman University |access-date=February 10, 2026}}</ref><ref>{{cite web |url=https://sites.google.com/view/andrewnjordan |title=Andrew N. Jordan |access-date=February 10, 2026}}</ref>

* [[Biography:Tenio Popmintchev|'''Tenio Popmintchev''']] extended phase-matched HHG into the soft and hard X-ray regions, demonstrating bright, coherent tabletop X-ray sources and pushing toward zeptosecond  resolution.<ref>{{cite journal |last1=Popmintchev |first1=Tenio |last2=Chen |first2=Ming-Chang |last3=Arpin |first3=Paul |last4=Murnane |first4=Margaret M. |last5=Kapteyn |first5=Henry C. |title=The attosecond nonlinear optics of bright coherent X-ray generation |journal=Nature Photonics |volume=4 |issue=12 |pages=822–832 |date=2010 |doi=10.1038/nphoton.2010.256}}</ref><ref>{{cite journal |last1=Popmintchev |first1=Tenio |last2=Chen |first2=Ming-Chang |last3=Popmintchev |first3=Dimitar |last4=Arpin |first4=Paul |last5=Brown |first5=Susannah |last6=Alix |first6=Carlos |last7=Bahabad |first7=Alon |last8=Murnane |first8=Margaret M. |last9=Kapteyn |first9=Henry C. |title=Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers |journal=Science |volume=336 |issue=6086 |pages=1287–1291 |date=2012 |doi=10.1126/science.1218497 |pmid=22679093}}</ref><ref>{{cite web |url=http://popmintchev.ucsd.edu/ |title=Popmintchev LABS – Quantum X-ray Science and Technology |publisher=University of California San Diego |access-date=February 10, 2026}}</ref><ref>{{cite web |url=https://profiles.ucsd.edu/tenio.popmintchev |title=Tenio Popmintchev |publisher=UC San Diego Profiles |access-date=February 10, 2026}}</ref>

* [[Biography:Dimitar Popmintchev|'''Dimitar Popmintchev''']] is a physicist at TU Wien's Photonics Institute, specializing in high-harmonic generation (HHG) for coherent EUV/soft X-ray sources.<ref>{{cite web |url=https://www.tuwien.at/en/etit/photonik/team |title=Team |publisher=TU Wien Institute of Photonics |access-date=February 10, 2026}}</ref><ref>{{cite web |url=https://orcid.org/0000-0003-4330-8235 |title=Dimitar Popmintchev (0000-0003-4330-8235) |publisher=ORCID |access-date=February 10, 2026}}</ref><ref>{{cite journal |last1=Popmintchev |first1=Tenio |last2=Chen |first2=Ming-Chang |last3=Popmintchev |first3=Dimitar |last4=Arpin |first4=Paul |last5=Brown |first5=Susannah |last6=Ališauskas |first6=Skirmantas |last7=Andriukaitis |first7=Giedrius |last8=Murnane |first8=Margaret M. |last9=Kapteyn |first9=Henry C. |title=Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers |journal=Science |volume=336 |issue=6086 |pages=1287–1291 |date=2012 |doi=10.1126/science.1218497 |pmid=22679093}}</ref><ref>{{cite web |url=https://scholar.google.com/citations?hl=en&user=DPFoVmMAAAAJ |title=Dimitar Popmintchev - Google Scholar |access-date=February 10, 2026}}</ref>
===Key contributions:===
* Efficient soft X-ray HHG in ionized plasmas (''Science'', 2015).<ref>{{cite journal |last=Popmintchev |first=Dimitar |last2=Chen |first2=Ming-Chang |last3=Arpin |first3=Paul |last4=Murnane |first4=Margaret M. |last5=Kapteyn |first5=Henry C. |title=Ultraviolet surprise: Efficient soft x-ray high-harmonic generation in multiply ionized plasmas |journal=Science |volume=350 |issue=6265 |pages=1225–1231 |date=2015 |doi=10.1126/science.aac9755}}</ref> 
* Resonant attosecond dynamics (''Phys. Rev. Research'', 2025).<ref>{{cite journal |last=Popmintchev |first=Dimitar |title=Attosecond Rabi oscillations in high harmonic generation resonantly driven by extreme ultraviolet laser fields |journal=Physical Review Research |volume=7 |issue=2 |pages=023268 |date=2025 |doi=10.1103/PhysRevResearch.7.023268}}</ref> 
* Tabletop zeptosecond X-ray sources (Popmintchev LABS).<ref>{{cite journal |last1=Hernández-García |first1=C. |last2=Popmintchev |first2=T. |last3=Murnane |first3=M. |last4=Kapteyn |first4=H. C. |title=Zeptosecond High Harmonic keV X-Ray Waveforms Driven by Midinfrared Laser Pulses |journal=Physical Review Letters |volume=111 |issue=3 |pages=033002 |date=2013 |doi=10.1103/PhysRevLett.111.033002}}</ref>These breakthroughs have revolutionised coherent X-ray science and the study of quantum dynamics in matter. Ongoing work by researchers like these continues to bridge foundational quantum principles with cutting-edge experiments and applications.

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[[File:Quantum Cosmology.jpg|130px|Quantum cosmology visualization]]
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== Cosmology ==
The equation <math>\hat{H}\Psi = 0</math> is known as the '''Wheeler–DeWitt equation''', a central result in the field of quantum cosmology. It is intended to describe the quantum state of the entire universe through a single wavefunction <math>\Psi</math>. Unlike standard quantum mechanical equations, the Wheeler–DeWitt equation is notable for its '''absence of an explicit time parameter'''. This reflects a deep conceptual issue in quantum gravity known as the “problem of time”: since the universe includes everything, there is no external clock against which evolution can be measured. As a result, the equation suggests that, at a fundamental level, the universe may be described as timeless, with all possible configurations encoded in the wavefunction.
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[[File:Schrodinger_equation light.jpg|250px]]
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In contrast, when a time parameter is introduced, the equation takes the familiar form of the '''time-dependent Schrödinger equation''', given by <math>i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi</math>. This equation governs the evolution of quantum systems in conventional quantum mechanics, where time is treated as an external parameter. The wavefunction <math>\Psi</math> evolves continuously with respect to time, and the Hamiltonian operator <math>\hat{H}</math> determines how this evolution proceeds.

The relationship between these two equations highlights a key distinction between ordinary quantum theory and attempts to formulate a quantum theory of gravity. While the Schrödinger equation assumes a fixed background time, the Wheeler–DeWitt equation removes this assumption, implying that time may instead be an emergent concept arising from correlations within the quantum state itself. In semiclassical approximations, it is possible to recover an effective notion of time from the Wheeler–DeWitt framework, allowing parts of the universe (such as matter fields) to evolve relative to others (such as the geometry of spacetime). This interplay between timeless fundamental laws and emergent temporal behavior remains one of the most extreme challenges in theoretical physics.

== Canonical quantum gravity ==
[[File:Quantum gravity combination.png]]
In [[Physics:Physics|physics]], '''canonical quantum gravity''' is an attempt to quantize the canonical formulation of general relativity (or '''canonical gravity'''). It is a [[Physics:Hamiltonian mechanics|Hamiltonian]] formulation of [[Biography:Albert Einstein|Einstein]]'s [[Physics:General relativity|general theory of relativity]]. The basic theory was outlined by [[Biography:Bryce DeWitt|Bryce DeWitt]]{{ref|dewitt}} in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann{{ref|bergmann}} using the so-called [[Physics:Canonical quantization|canonical quantization]] techniques for constrained Hamiltonian systems invented by [[Biography:Paul Dirac|Paul Dirac]].{{ref|dirac}} Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed [[Physics:Gauge fixing|gauge choice]]. Newer approaches based in part on the work of DeWitt and Dirac include the [[Astronomy:Hartle–Hawking state|Hartle–Hawking state]], [[Physics:Regge calculus|Regge calculus]], the [[Wheeler–DeWitt equation]] and [[Physics:Loop quantum gravity|loop quantum gravity]].

== Core concepts and introduction to the quantum world ==
*[[Physics:Quantum|'''Quantum''']]: The smallest unit of a physical property. Imagine a single “portion” of energy or matter that cannot be broken down any further. <br>[https://www.fisica.net/mecanica-quantica/Griffiths%20-%20Introduction%20to%20quantum%20mechanics.pdf For introduction to Quantum Mechanics (QM) and learning about its behaviour this course is highly recommended]
* [https://oyc.yale.edu/physics/phys-201/lecture-24#:~:text=Overview,of%20definite%20energy)%20is%20discussed. Above lecture does not include Schödingers Time Dependant formula, if you follow this lecture it will show you a formula to predict the future.]
* [[Physics:Physics|'''Physics''']] is the [[Wikipedia:scientific|scientific]] study of [[Physics:Matter|matter]], its [[Physics:Elementary particle|fundamental constituents]], its [[Physics:Motion|motion]] and behavior through [[Physics:Space physics|space]] and [[Wikipedia:time|time]], and the related entities of [[Wikipedia:energy|energy]] and [[Wikipedia:force|force]].<ref name="maxwell1878-physicalscience">{{harvnb|Maxwell|1878|p=9}} "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events."</ref> It is one of the most fundamental scientific disciplines.<ref name="youngfreedman2014p1">{{harvnb |Young|Freedman|2014|p=1}} "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."</ref><ref name="youngfreedman2014p2">{{harvnb |Young|Freedman|2014|p=2}} "Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena."</ref><ref name="holzner2003-physics">{{harvnb|Holzner|2006|p=7}} "Physics is the study of your world and the world and universe around you."</ref>
* '''Quantum Physics (QP)''' studies energy and matter at the fundamental level. To reveal the behaviors and properties of the building blocks of the universe. Quantum experiments involve very small objects, named electrons and photons, quantum phenomena are everywhere,&nbsp; on every scale.
[[file:Wignerfunction_fock_4.png|thumb|Wigner function for a Fock state with photon number n = 4 (shows structure for non-vacuum Fock states).]]
* [[Wikipedia:Quantum field theory|'''Quantum field theory''']], here the state of the system is described by a state vector in [[Wikipedia:Fock space|Fock space]], which includes states with different numbers of particles. The vacuum state (no particles) is denoted <math>|0\rangle</math>, and satisfies <math>
a_p |0\rangle = 0 \quad \text{for all momenta } p
</math> (where <math>a_p</math> is the annihilation operator for mode <math>p</math>). The quantum field <math>\phi(x,t)</math> is an operator that can create or destroy particles. It is built from creation and annihilation operators:<math display="block">\phi(x,t) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_p}} \left[ a_p \, e^{-i p \cdot x} + a_p^\dagger \, e^{i p \cdot x} \right]</math>(with appropriate relativistic conventions for the 4-vector <math>p \cdot x = \omega_p t - \mathbf{p} \cdot \mathbf{x}</math> and <math>\omega_p = \sqrt{\mathbf{p}^2 + m^2}</math> for a scalar field).Its action on the vacuum creates a superposition of one-particle states:<math>\phi(x,t) |0\rangle \ \propto\ </math> a "smeared" one-particle state peaked around position <math>x</math>(this is sometimes loosely called "creating a particle at <math>x</math>", but remember it is not sharply localized). On a single-particle state <math>|\psi\rangle</math> (say with wavefunction <math>\psi(p)</math> in momentum space), the field can annihilate the particle (giving vacuum) or create an extra one (giving two particles). Roughly speaking, the annihilation part of the field acts like:<math> [\text{annihilation part of } \phi(x)] \ |\text{particle at/near } y\rangle \ \approx \ \delta(x-y) \, |0\rangle
</math> This is why field operators are said to "probe" for particles at a point — but always remember the full <math>\phi</math> contains both creation and annihilation parts.
*[[Wikipedia:Quantum mechanics|'''Quantum Mechanics''']]: can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales.
[[File:Wave-particle.svg|thumb|Wave-Particle Duality]]
* [[Wikipedia:Wave–particle duality|'''Wave–particle duality''']]: Particles like electrons sometimes act like solid objects and sometimes like waves. The famous double-slit experiment shows this: electrons send through two narrow openings create an interference pattern, behaving like waves, but when measured, they appear as particles. The observation itself seems to influence reality.

== Superposition and quantum entanglement ==
[[File:Quantum superposition.svg|thumb|Quantum Superposition and Probability Density]]
* [[Wikipedia:Superposition|'''Superposition:''']] A quantum particle can exist in several different states simultaneously. Schrödinger captured this strangeness with his famous thought experiment: a cat sealed in a box, tied to a random quantum event. Until the box is opened and observed, the cat is, in a very real quantum sense, both alive and dead at the same time.  This dramatically illustrates how quantum logic defies our everyday intuition.
[[File:Entanglement filtering via anti-parity time (APT) symmetry.png|thumb|Entanglement via  (APT)]]

* [[Wikipedia:Quantum entanglement|'''Quantum entanglement''':]] Two or more particles can become linked so that the state of one influences the other, even at a distance. Einstein called this “spooky action at a distance.” Experiments have confirmed it, as the basis of quantum computing and secure communication.<br><br>

===Theoretical Perspective on Distance===
In quantum theory, there is '''no fundamental limit''' to the distance over which particles can remain entangled. Once entangled, particles stay correlated indefinitely, as long as their quantum states are not disturbed by interactions with the environment (a process called decoherence). This is true even across large distances, since entanglement is a property of the quantum wave function and doesn't rely on signals traveling between the particles, it's instantaneous and non-local. The key point is that no information can be transmitted faster than light via entanglement (due to the no-communication theorem), preserving causality in relativity. In principle, entanglement could span the entire universe, limited only by practical challenges like maintaining coherence.

===Practical Limits and Experimental Distances===
[[File:Artist impression of Quantum internet.png|thumb|Artist impression of Quantum internet]]
While theory allows unlimited distance, real-world experiments face limitations from decoherence (e.g., due to interactions with air molecules, fiber optic losses, or noise). Photons (light particles) are commonly used for long-distance tests because they travel far with minimal interaction, often via optical fibers, free space, or satellites. Here's a summary of key achievements:

* '''Short to Medium Distances (Lab-Scale)''': Early tests, like Alain Aspect's 1982 experiment, confirmed entanglement over meters. More recent ion-based entanglement has reached 230 meters in labs, useful for quantum computing prototypes. Atomic entanglement over 33 km of fiber optic cable was achieved in 2022, setting a record for matter-based systems.

* '''Long-Distance Ground-Based''': In 2022, researchers entangled photons over 248 km of optical fiber, a new record for fiber-based transmission. This doubled the previous 100 km mark and is a step toward practical quantum networks. Atmospheric tests have reached 144 km (e.g., between Canary Islands in 2007).

* '''Space-Based and Satellite Records''': The farthest confirmed entanglement is about 1,203 km, achieved in 2017 using China's Micius satellite. Photons were beamed from space to ground stations in China, violating Bell inequalities (a test for true entanglement) and demonstrating survival over extreme distances. This shattered earlier records like 88 miles (142 km) in 2012.

* '''Even Heavier Particles''': In 2024, the CMS experiment at CERN observed entanglement between top quarks (the heaviest known particles) at the Large Hadron Collider. While the physical distance is tiny (subatomic), their high relative speeds mean the effective "distance" for any classical signal would exceed light-speed limits, confirming non-local action.

These distances are pushing toward a "[https://www.quantuminternet.it/ quantum internet]," where entanglement enables secure communication (e.g., [[Wikipedia:quantum key distribution|quantum key distribution]]) over global scales. Current records are around 1,200 km for photons, but ongoing work (e.g., with quantum repeaters) aims for intercontinental links.

== Uncertainty Principle ==
[[File:QHO-coherentstate3-animation-color.gif|thumb|250px|QHO coherent state (animation)]]
[[File:qho_narrow.gif|thumb|250px|Position and momentum of a Gaussian initial state for a QHO, wide]]
[[File:Heisenberg's Microscope 2.gif|thumb|250px|Illustration of Heisenberg's gamma-ray microscope thought experiment: high-resolution light disturbs momentum. Source: Wikimedia Commons.]]

[[File:Heisenberg microscope with wavefronts and electron scatter.svg|thumb|250px|Heisenberg microscope with wavefronts and electron scatter]]

This inequality means the product of the uncertainties is '''at least''' a small but non-zero constant, <math>\Delta x</math> and <math>\Delta p</math> can not be arbitrarily small at the same time.

The following images illustrate how a narrow position uncertainty forces a large spread in momentum (and vice versa), shown with Gaussian wave packets:

This is '''not''' a limitation of our measuring instruments, it is a fundamental property of nature. Particles do not have definite position '''and''' momentum simultaneously in the classical sense; their quantum state is described by a wave function that inherently spreads out these properties.

A thought experiment illustrating this is [[Wikipedia:Heisenberg's microscope|'''Heisenberg's microscope''']]: To pinpoint a particle's position very accurately, you need short-wavelength (high-energy) light, but that high-energy photon kicks the particle hard, disturbing its momentum wildly. Longer-wavelength light is gentler but gives blurry position information.

'''The uncertainty principle:''' applies to other conjugate pairs as well. A commonly cited example is energy and time, <math>\Delta E \cdot \Delta t \gtrsim \frac{\hbar}{2}</math>, which reflects a trade-off between the precision of energy measurements and the characteristic timescale of a process. This helps explain phenomena such as virtual particles, quantum tunneling, and the stability of atoms: electrons cannot be localized arbitrarily close to the nucleus without acquiring a large momentum uncertainty.

The quantum world is not simply "small classical physics"; it has built-in fuzziness and trade-offs that challenge our intuitions about reality.

<math>\hbar</math> is very small (about <math>10^{-34}</math> J·s), so these uncertainties are negligible for macroscopic objects. This is why classical physics works so well for everyday systems like baseballs and cars. At the atomic scale, however, this limit shapes everything and underlies the probabilistic nature of quantum mechanics: perfect certainty is impossible, and physical systems are described in terms of probability distributions rather than definite trajectories.

== The Quantum Revolution ==

For centuries, scientists saw the universe as a precise, predictable apparatus. [[Wikipedia:Classical mechanics|Newtonian physics]] suggested that if you knew the position and motion of every object, you could predict the future perfectly. But the early twentieth century shattered this view. Experiments with atoms and light revealed that classical laws no longer applied. Energy was quantized, light sometimes behaved like a wave and sometimes like particles, and certainty gave way to probability.

Quantum Mechanics emerged as a revolutionary framework, changing how we understand nature. Unlike classical physics, it does not give precise predictions. Instead, it provides probabilities, the propability of finding a particle in a particular place at a particular time. This uncertainty is not a flaw; it is the fundament of reality.

== The Weirdness That Works ==

Though abstract, Quantum Mechanics touch our daily lives. Semiconductors in computers, lasers in medical devices, and the atomic clocks that power GPS systems all rely on quantum principles. Superposition and entanglement are no longer just theoretical oddities,  quantum computing, promising machines that solve certain problems far faster than any classical computer. Quantum communication uses entanglement to create ultra-secure information channels resistant to eavesdropping.

Even as quantum theory transforms technology, it challenges our intuition. Cause and effect, certainty, and even the very notions of space and time seem different at the quantum level. Particles can appear in two places at once, outcomes can seem random, and yet an underlying order governs it all.

== Looking Into Reality ==

Quantum Mechanics challenges our way of thinking about reality itself. While the world we see follows simple, predictable rules, the universe at its smallest scales is based on principles that seem almost unreal. It is a place where observation shapes reality, where distant particles are linked, and where probabilities replace certainties.

Exploring quantum is not just about physics; it is traveling into the nature of the universe. Every discovery, from quantum computers to secure communication systems, shows a world more intricate than our everyday experience suggests.

The quantum world may not be intuitive, but it shows the elegance of nature.With its rules, new technologies evolve, and a deeper understanding of reality, a universe more subtle  than we ever imagined.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
[[Category:Quantum mechanics]]
[[Category:Physics]]

=References=
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Quantum basics|1}}