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{{Short description|Scientific overview of quantum mechanics sizes}}
{{Quantum book backlink|Mathematical structure and systems}}
 
'''Approach to Mesoscopic Physics: Quantum Size Effects'''
'''Mesoscopic physics''' is a field within [[Wikipedia:condensed matter physics|condensed matter physics]] that describes systems whose dimensions are intermediate between the [[Wikipedia:nanoscopic scale|nanoscale]] of individual [[Wikipedia:atom|atoms]] or [[Wikipedia:molecule|molecules]] and the [[Wikipedia:micrometre|micrometre]] scale of bulk materials.<ref>{{cite journal |last1=Muller|first1=M. |last2=Katsov|first2=K. |last3=Schick|first3=M. |date=November 2006 |title=Biological and synthetic membranes: What can be learned from a coarse-grained description? |journal=Physics Reports |volume=434 |issue=5–6 |pages=113–176 |doi=10.1016/j.physrep.2006.08.003 |issn=0370-1573 |arxiv=cond-mat/0609295 |bibcode=2006PhR...434..113M |s2cid=16012275}}</ref> Systems large enough to contain many atoms, yet still small enough so that the motion of particles is influenced by [[Wikipedia:quantum mechanics|quantum mechanical]] effects rather than being fully described by [[Wikipedia:classical mechanics|classical physics]].<ref name=Meso-1>{{cite encyclopedia |title=Sci-Tech Dictionary |encyclopedia=McGraw-Hill Dictionary of Scientific and Technical Terms |year=2003 |publisher=[[Wikipedia:S&P Global|McGraw-Hill Companies, Inc.]]}}</ref><ref name=Meso-2/>

[[File:Scale of the universe.jpg|thumb|450px|Take a visual tour across scales, seeing how quantum phenomena appear at nature’s small levels. This infographic shows the territory where classical physics submits to quantum behavior, everything from everyday objects to atoms to subatomic particles and down into the Planck scale. It exposes a world dictated by probabilities, wave functions and limits of measurement.]]
==Objective==
[[File:Artistic impression of an atom 2a.jpg|thumb|Artistic impression of an atom 2a]]
{{condensed matter physics}}
This article is based on Wikipedia articles and other sources.The objective is to explain how "sizes" are used in Quantum Mechanics (QM). Often sizes in quantum mechanics are probabilistic, with particles not having a fixed size, but a size that depends on factors like wavelength or boundary condition.

An [[Wikipedia:electronic device|electronic device]] miniaturized from [[Wikipedia:Macroscopic scale|macroscopic]] to mesoscopic dimensions (see also [[Wikipedia:Carlo Beenakker|Carlo Beenakker]]) behaves differently as a consequence of [[Wikipedia:Coherence (physics)|quantum coherence]]. Macroscopic wire shows a smooth increase in [[Wikipedia:electrical conductance|electrical conductance]] with thickness, but a wire at [[Wikipedia:Length scale|mesoscopic scale]] exhibits [[Wikipedia:quantization (physics)|quantized]] [[Wikipedia:Conductance quantum|conductance]], increasing in discrete steps rather than continuously. Research in this area uses both experimental measurements and [[Wikipedia:theoretical|theoretical models]] to explore transport phenomena in [[Wikipedia:insulator (electrical)|insulators]], [[Wikipedia:semiconductor|semiconductors]], [[Wikipedia:metal|metals]], and [[Wikipedia:superconductor|superconductors]]. The field also has applications in the engineering of [[Wikipedia:Nanotechnology|nanoscale electronic components]].<ref name=Meso-1/><ref name=Meso-2/> Important for mesoscopic physics is that the physical properties of materials—mechanical, chemical, and electrical—change significantly during miniaturization. Objects shrink toward the [[Wikipedia:nanoscopic scale|nanoscale]], the fraction of atoms located at the surface becomes large enough to influence behavior. In contrast, for conventional bulk materials larger than about one micrometre, surface effects are negligible compared to the total number of atoms. Mesoscopic research focuses on metallic or semiconducting structures produced using [[Wikipedia:nanofabrication|nanofabrication]] and [[Wikipedia:microelectronic|microelectronic]] techniques.<ref name=Meso-1/><ref name=Meso-2/>

There is no single precise size boundary that defines a mesoscopic system; it typically refers to structures ranging from about 100 nm (the size of a small [[Wikipedia:virus|virus]]) up to around 1,000 nm (the size of a [[Wikipedia:bacterium|bacterium]]), with 100 nm often regarded as the upper size limit for a [[Wikipedia:nanoparticle|nanoparticle]]. Mesoscopic physics is related to [[nanotechnology]] and nanoscale device engineering. Three common types of phenomena in mesoscopic systems include [[Wikipedia:SQUID|quantum interference]], [[Wikipedia:Potential well|quantum confinement]], and [[Wikipedia:Elementary charge|electron charging effects]].<ref name="Meso-1" /><ref name="Meso-2">"Mesoscopic physics." McGraw-Hill Encyclopedia of Science and Technology. The McGraw-Hill Companies, Inc., 2005. Answers.com 25 Jan 2010. http://www.answers.com/topic/mesoscopic-physics-1
</ref>

== Research on sizes ==
[[File:62cts Brazilian Crystal Opal.JPG|thumb|Brazilian Crystal Opal.]] ⋅'''[[Wikipedia:Alexey Ekimov|Alexey Ekimov]]''' or '''Aleksey Yekimov''' [[Wikipedia:Solid-state physics|solid state physicist]] and a pioneer in [[Wikipedia:nanomaterials|nanomaterials]] research. He discovered the [[Wikipedia:semiconductor|semiconductor]] nanocrystals known as [[Wikipedia:quantum dot|quantum dots]] in 1981, while working at the [[Wikipedia:Vavilov State Optical Institute|Vavilov State Optical Institute]].<ref>{{cite journal |author=Екимов А.И., Онущенко А.А. |title=Квантовый размерный эффект в трехмерных микрокристаллах полупроводников |journal=Письма в ЖЭТФ |volume=34 |pages=363–366 |year=1981 |url=https://link.springer.com/article/10.1134/S0021364015230034 |access-date=26 March 2015 |archive-date=16 December 2014 |archive-url=https://web.archive.org/web/20141216142832/http://www.jetpletters.ac.ru/ps/1030/article_15644.pdf |url-status=dead }}</ref><ref>{{Cite web |date=2023-10-04 |title=Russian-Born Quantum Dot Pioneer Ekimov Wins Nobel Prize in Chemistry |url=https://www.themoscowtimes.com/2023/10/04/russian-born-quantum-dot-pioneer-ekimov-wins-nobel-prize-in-chemistry-a82660 |access-date=2023-10-06 |website=The Moscow Times |language=en}}</ref><ref name=":0">{{Cite web |date=2023-10-09 |title=Alexei Ekimov {{!}} Biography, Nobel Prize, Quantum Dots, & Facts {{!}} Britannica |url=https://www.britannica.com/biography/Alexei-Ekimov |access-date=2023-10-11 |website=www.britannica.com |language=en}}</ref> In 2023, he was awarded the [[Wikipedia:Nobel Prize in Chemistry|Nobel Prize in Chemistry]] for this discovery.

In 1981, Ekimov, along with Alexei A. Onushchenko <ref name="Ekimov1981">{{cite journal |last=Ekimov |first=A. I. |last2=Onushchenko |first2=A. A. |title=Quantum size effect in three-dimensional microscopic semiconductor crystals |url=https://link.springer.com/article/10.1134/S0021364023130040|journal=JETP Letters |volume=34 |issue=6 |pages=345–349 |year=1981}}</ref> reported the discovery of [[Wikipedia:quantum size effects|quantum size effects]] in [[Wikipedia:Copper(I) chloride|copper chloride]] nanocrystals in glass,<ref name="Sanderson">{{cite journal |last=Sanderson |first=Katharine |last2=Castelvecchi |first2=Davide |title=Tiny 'quantum dot' particles win chemistry Nobel |journal=Nature |date=4 October 2023 |volume=622 |issue=7982 |pages=227–228 |doi=10.1038/d41586-023-03048-9 |pmid=37794149}}</ref><ref name="Gramling">{{cite news |last1=Gramling |first1=Carolyn |title=The development of quantum dots wins the 2023 Nobel prize in chemistry |url=https://www.sciencenews.org/article/quantum-dots-nanoparticles-bawendi-brus-ekimov |work=Science News |date=4 October 2023}}</ref><ref name="Clery">{{cite news |last1=Clery |first1=Daniel |last2=Kean |first2=Sam |title=Creators of quantum dots, used in TV displays and cell studies, win chemistry Nobel |url=https://www.science.org/content/article/creators-of-quantum-dots-used-in-tv-displays-and-cell-studies-win-chemistry-nobel |work=Science |date=4 Oct 2023}}</ref><ref>{{cite journal |last1=Ekimov |first1=A. I. |last2=Onushchenko |first2=A. A. |date=1981-09-01 |title=Quantum size effect in three-dimensional microscopic semiconductor crystals |journal=Soviet Journal of Experimental and Theoretical Physics Letters |volume=34 |pages=345–349}}</ref> a phenomenon now known as [[Wikipedia:quantum dots|quantum dots]]. During his time at the institute he further investigated these system and developed the theory of [[Wikipedia:quantum confinement|quantum confinement]] with [[Wikipedia:Alexander Efros|Alexander Efros]].<ref>{{cite journal |last1=Efros |first1=Alexander L. |last2=Brus |first2=Louis E. |title=Nanocrystal Quantum Dots: From Discovery to Modern Development |journal=ACS Nano |date=27 April 2021 |volume=15 |issue=4 |pages=6192–6210 |doi=10.1021/acsnano.1c01399 |pmid=33830732}}</ref><ref>{{cite journal |last1=Ekimov |first1=A. I. |last2=Onushchenko |first2=A. A. |last3=Plyukhin |first3=A. G. |last4=Efros |first4=Al. L. |date=1985-04-01 |title=Size quantization of excitons and determination of the parameters of their energy spectrum in CuCl |journal=Physical Review B |volume=31 |issue=2 |pages=1021–1026 |url=https://www.researchgate.net/publication/285831802_Size_quantization_of_exciton_and_determination_of_their_energy_spectrum_parameters_in_CuCl}}</ref>

== Quantum Tunneling ==
[[File:Quantum-tunneling.svg|thumb|Quantum-tunneling]]
[[Wikipedia:Quantum tunnelling|Tunneling]] is directly related to the wave nature of matter.<ref>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |edition=2nd |publisher=Pearson Prentice Hall |year=2005 |isbn=978-0131118928}}</ref> Quantum tunneling is a quantum mechanical phenomenon in which particles, such as electrons or protons as [[Wikipedia:Wave packet|wave packets]], can pass through [[Wikipedia:Rectangular potential barrier|potential energy barriers]] even when they do not have enough classical energy to overcome them.<ref>{{cite journal |last=Gamow |first=G. |title=Zur Quantentheorie des Atomkernes |journal=Zeitschrift für Physik |volume=51 |pages=204–212 |year=1928 |doi=10.1007/BF01343196}}</ref><ref>{{cite journal |last=Fowler |first=R. H. |last2=Nordheim |first2=L. |title=Electron emission in intense electric fields |journal=Proceedings of the Royal Society A |volume=119 |pages=173–181 |year=1928 |doi=10.1098/rspa.1928.0091}}</ref> In classical mechanics, this would be impossible. Low-mass particles are most likely to tunnel, and the probability decreases rapidly with increasing particle mass or barrier width.<ref>{{cite book |last=Cohen-Tannoudji |first=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Frank |title=Quantum Mechanics |publisher=Wiley |year=1977 |isbn=978-0471164333}}</ref> For electrons, tunneling can be significant through barriers with thicknesses of about 1–3 nm, while for protons or hydrogen atoms, it is typically only observable for much thinner barriers, around ≤0.1 nm.<ref>{{cite journal |last=Hänggi |first=P. |last2=Talkner |first2=P. |last3=Borkovec |first3=M. |title=Reaction-rate theory: fifty years after Kramers |journal=Reviews of Modern Physics |volume=62 |issue=2 |pages=251–341 |year=1990 |doi=10.1103/RevModPhys.62.251}}</ref> The principle of tunneling leads to the development of [[Wikipedia:Scanning tunneling microscope|Scanning Tunneling Microscope (STM)]] which had a serious impact on chemical, biological and material science research.<ref>{{cite journal |last=Binnig |first=G. |last2=Rohrer |first2=H. |title=Scanning tunneling microscopy |journal=Surface Science |volume=126 |pages=236–244 |year=1983 |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.57}}</ref>

== Planck length ==
[[File:Triangle of everything simplified 2 triangle of everything - Planck Units.png | thumb|upright=0.8 | A mass–radius log plot of various objects]]
The [[Wikipedia:Planck length|Planck length]](<math>\ell_P</math>) is often considered the smallest meaningful length in physics, obtained by combining quantum mechanics, relativity, and gravity:
<math>
\ell_P = \sqrt{\frac{\hbar G}{c^3}} \approx 1.616 \times 10^{-35} \text{ m}.
</math>
At this scale, quantum fluctuations of spacetime are expected to become significant, and both ordinary quantum mechanics and general relativity break down. A complete theory of [[Wikipedia:quantum gravity|quantum gravity]] (such as [[Wikipedia:string theory|string theory]] or [[Wikipedia:loop quantum gravity|loop quantum gravity]]) would be required to describe physics below this length.

==Schwarzschild radius==
The '''Schwarzschild radius''' is a parameter in the [[Wikipedia:Schwarzschild solution|Schwarzschild solution]] to [[Wikipedia:Einstein's field equation|Einstein's field equation]]s that corresponds to the [[Wikipedia:radius|radius]] of a sphere in flat space that has the same surface area as that of the [[Wikipedia:event horizon|event horizon]] of a Schwarzschild [[Wikipedia:black hole|black hole]] of a given mass. It is a characteristic quantity that may be associated with any quantity of mass. The Schwarzschild radius was named after the German astronomer [[Wikipedia:Karl Schwarzschild|Karl Schwarzschild]], who calculated this solution for the theory of [[Wikipedia:general relativity|general relativity]] in 1916.

The Schwarzschild radius is given as
<math display="block"> r_\text{s} = \frac{2 G M}{c^2} ,</math>
where ''G'' is the [[Wikipedia:Newtonian constant of gravitation|Newtonian constant of gravitation]], ''M'' is the mass of the object, and ''c'' is the [[Wikipedia:speed of light|speed of light]].<ref>{{cite book |last=Kutner |first=Marc Leslie |url=https://archive.org/details/astronomyphysica00kutn/ |title=Astronomy: a physical perspective |date=2003 |publisher=Cambridge University Press |isbn=978-0-521-82196-4 |edition=2nd |location=Cambridge, U.K.; New York |pages=148}}</ref><ref>{{Cite book |last=Guidry |first=M. W. |title=Modern general relativity: black holes, gravitational waves, and cosmology |date=2019 |publisher=Cambridge University Press |isbn=978-1-107-19789-3 |location=Cambridge; New York, NY |pages=92}}</ref>

== Compton wavelength ==
[[File:CGRO s37-96-010.jpg|thumb|150px|The Compton Gamma Ray Observatory]]
For a particle of mass <math>m</math>, the [[Wikipedia:Compton wavelength|Compton wavelength]] (<math>\lambda_C</math>) is defined as <math> \lambda_C = \frac{h}{m c}.
</math>
It represents the smallest region in which a particle can be localized without creating particle–antiparticle pairs. For example:

*Electron: <math>\lambda_C \approx 2.4 \times 10^{-12}</math> m

*Proton: <math>\lambda_C \approx 1.3 \times 10^{-15}</math> m

== de Broglie wavelength ==
The [[Wikipedia:de Broglie wavelength|de Broglie wavelength]] (<math>\lambda</math>) depends on the particle’s momentum <math>p</math>:
<math>
\lambda = \frac{h}{p}.
</math>
This wavelength can be made arbitrarily small by increasing the particle’s momentum, so there is no fixed minimum scale in [[Wikipedia:Quantum gravity|non-gravitational quantum mechanics]].
The wavelength of a sine wave, λ, is measured between two points ot the same phase, between crests (on top), or troughs (on bottom), or zero crossings as shown.

== Experimental limits ==
[[File:CMS Higgs-event.jpg|thumb|90px|LHC<math>10^{-19}</math>]]
Experiments have probed distances down to about <math>10^{-19}</math> m (the scale of high-energy collisions at the [[Wikipedia:Large Hadron Collider|Large Hadron Collider]]), and elementary particles still appear pointlike at these scales.

== Limits of measurement below the Planck length ==
[[File:Quantum mechanics travelling wavefunctions wavelength.svg|thumb|Position x and momentum p wavefunctions corresponding to quantum particles.]]
Attempts to measure distances smaller than the Planck length encounter fundamental limits due to the combination of quantum mechanics and general relativity.

According to the [[Wikipedia:Heisenberg uncertainty principle|Heisenberg uncertainty principle]]:
<math>
\Delta x , \Delta p \gtrsim \frac{\hbar}{2}.
</math> To probe a very small region <math>\Delta x</math>, a particle with very large momentum <math>p</math> (and hence very high energy <math>E \approx pc</math>) is required.

However, according to general relativity, concentrating too much energy into a small region of space will create a black hole if the energy corresponds to a [[Wikipedia:Schwarzschild radius|Schwarzschild radius]] <math> r_s = \frac{2 G E}{c^4}.
</math>When the Schwarzschild radius becomes comparable to the uncertainty in position (<math>r_s \approx \Delta x</math>), further localization becomes impossible, because the region collapses into a black hole.

Combining these relations gives an approximate limit: <math>
\Delta x \gtrsim \sqrt{\frac{\hbar G}{c^3}} = \ell_P.
</math> Thus, the Planck length represents the smallest measurable distance in principle: below this scale, the very concept of "position" loses operational meaning. <ref>{{cite journal |last=Mead |first=C. A. |title=Possible Connection Between Gravitation and Fundamental Length |journal=Physical Review |volume=135 |issue=3B |pages=B849–B862 |year=1964 |doi=10.1103/PhysRev.135.B849}}</ref><ref>{{cite journal
| last=Garay
| first=L. J.
| title=Quantum gravity and minimum length
| journal=International Journal of Modern Physics A
| volume=10
| issue=2
| pages=145–165
| year=1995
| doi=10.1142/S0217751X95000085
}}</ref><ref>{{cite journal |last=Ng |first=Y. J. |last2=van Dam |first2=H. |title=Limit to space-time measurement |journal=Modern Physics Letters A |volume=9 |issue=4 |pages=335–340 |year=1994 |url=https://www.worldscientific.com/doi/abs/10.1142/S0217732394000356}}</ref> 

== Conceptual implications ==
The existence of a minimum length scale is a common feature in approaches to quantum gravity, including string theory and loop quantum gravity. In these theories, spacetime may have a discrete or quantized structure at the Planck scale, preventing the definition of smaller distances.<ref>{{cite book |last=Rovelli |first=C. |title=Quantum Gravity |publisher=Cambridge University Press |year=2004 |isbn=978-0521837333}}</ref><ref>{{cite book |last=Thiemann |first=T. |title=Modern Canonical Quantum General Relativity |publisher=Cambridge University Press |year=2007 |url=https://arxiv.org/abs/gr-qc/0110034}}</ref><ref>{{cite book |last=Polchinski |first=J. |title=String Theory, Vols. 1 & 2 |publisher=Cambridge University Press |year=1998 |url=https://www.cambridge.org/core/books/string-theory/30409AF2BDE27D53E275FDA395AB667A}}</ref><ref>{{cite journal |last=Hossenfelder |first=S. |title=Minimal Length Scale Scenarios for Quantum Gravity |journal=Living Reviews in Relativity |volume=16 |pages=2 |year=2013 |doi=10.12942/lrr-2013-2}}</ref>

== Summary ==
Meaning -Planck length, Electron, Compton wavelength, de Broglie wavelength- Quantum mechanics itself does not impose a fundamental smallest size, but when gravity is included, the ''Planck length'' is often regarded as the smallest physically meaningful scale. 
*Size and Quantum Effects: Quantum mechanics becomes significant at scales on the order of nanometers (10⁻⁹ meters) or smaller, where properties like wave-particle duality, superposition, and quantization of energy levels dominate. For example, in atoms, electrons occupy discrete energy levels determined by the size of their orbitals.

*Scaling and Classical Transition: As size increases to macroscopic scales, quantum effects become negligible due to decoherence and the averaging of quantum probabilities. This is why classical mechanics describes larger systems effectively.

=Specific Contexts=
* Quantum Confinement: In nanostructures like quantum dots, the physical size of the system restricts electron movement, leading to quantized energy levels that depend directly on size.
* Heisenberg Uncertainty Principle: The smaller the spatial confinement (size), the larger the uncertainty in momentum, a purely quantum effect.
* Macroscopic Quantum Phenomena: In rare cases, like superconductors or Bose-Einstein condensates, quantum effects persist at larger scales, but these are exceptions and often involve low temperatures or specific conditions.

== Theory ==
{{Quantum mechanics}}
A quantum (plural quanta) is the smallest discrete unit of a physical property, such as energy, light, or angular momentum. For example, a [[Wikipedia:photon|photon]] is a quantum of light.

*[[Wikipedia:Quantum physics|Quantum physics]] is the branch of science that studies the behavior of matter and energy at very small scales, such as atoms and subatomic particles. It explores phenomena that classical physics cannot explain, including [[Wikipedia:wave-particle duality|wave-particle duality]] and quantized energy levels.

*[[Wikipedia:Quantum mechanics|Quantum mechanics]] is the mathematical framework within quantum physics that provides the rules and equations to describe and predict the behavior of [[Wikipedia:quantum systems|quantum systems]]. It includes principles such as the [[Wikipedia:uncertainty principle|uncertainty principle]], [[Wikipedia:wavefunctions|wavefunctions]], and [[Wikipedia:superposition|superposition]].

So:
* Quantum = the smallest piece of a property.
* Quantum physics = the study of the behavior of these small pieces.
* Quantum mechanics = the set of rules and equations that describe how they behave.

Quantum Science consist of Quantum physics (QP) and Quantum mechanics (QM) describing the behaviour of matter and light at the atomic and subatomic scale.<ref name="Griffiths2018">{{Cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |publisher=Cambridge University Press |year=2018 |isbn=978-1107189638 |edition=3rd |location=Cambridge}}</ref> These phenomena underlie technologies such as [[Wikipedia:Semiconductor|semiconductors]], [[Wikipedia:Laser|lasers]], and [[Wikipedia:Solar cell|solar cells]], and form the basis of developing fields including [[Quantum computing|quantum computing]] and [[Wikipedia:Quantum sensor|quantum sensing]].<ref name="DowlingMilburn2003">{{Cite journal |last1=Dowling |first1=Jonathan P. |last2=Milburn |first2=Gerard J. |year=2003 |title=Quantum technology: the second quantum revolution |url=https://royalsocietypublishing.org/doi/10.1098/rsta.2003.1227 |journal=Philosophical Transactions of the Royal Society A |volume=361 |issue=1809 |pages=1655–1674 |doi=10.1098/rsta.2003.1227 |pmid=12952679 |bibcode=2003RSPTA.361.1655D }}</ref>

=See also=
[[File:Comparison of nanomaterials sizes.jpg|thumb|Comparison of nanomaterials sizes]]
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
[[Category:Quantum mechanics]]
[[Category:Physics]]

== References ==
{{Reflist|3}}

{{Author|Harold Foppele}}
[[Category:Quantum mechanics]]
[[Category:Physics]]

{{Sourceattribution|Quantum A Matter Of Size|1}}

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imported>WikiHarold: Repair Quantum Collection B backlink template

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