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{{Short description|Minimum amount of a physical entity involved in an interaction}}
The word '''quantum''', pl. "quanta", comes from the [[Latin]] "quantus", for "how much". In general, it refers to an "amount of something". But, the term is often used in the more specific sense which it has in [[physics]], where a quantum refers to an indivisible, and perhaps elementary entity. For instance, a "[[light quantum]]", being a unit of light (that is, a [[photon]]). In combinations like "[[quantum mechanics]]", "[[quantum optics]]", etc., it distinguishes a more specialized field of study.
{{Other uses|Quantum (disambiguation)}}


In [[physics]], a '''quantum''' ({{plural form}}: '''quanta''') is the minimum amount of any physical entity ([[physical property]]) involved in an [[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 property can be "quantized" is referred to as "the hypothesis of [[quantization (physics)|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 [[Wiktionary:discrete|discrete]] values consisting of [[Multiple (mathematics)|integer multiples]] of one quantum. For example, a [[photon]] is a single quantum of [[light]] of a specific [[frequency]] (or of any other form of [[electromagnetic radiation]]). Similarly, the energy of an [[electron]] bound within an [[atom]] is quantized and can exist only in certain discrete values.<ref>{{Cite book |last=Rovelli |first=Carlo |title=Reality is not what it seems: the elementary structure of things |date=January 2017 |publisher=Riverhead Books |isbn=978-0-7352-1392-0 |edition=1st American |location=New York, New York |pages=109–130 |translator-last=Carnell |translator-first=Simon |translator-last2=Segre |translator-first2=Erica}}</ref> 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 [[quantum mechanics]]. Quantization of [[energy]] and its influence on how energy and matter interact ([[quantum electrodynamics]]) is part of the fundamental framework for understanding and describing nature.
Behind this, one finds the fundamental notion that a physical property may be "quantized", referred to as "[[quantization (physics)|quantization]]". This means that the magnitude can take on only certain [[numerical]] values, rather than any value, at least within a range. For example, the energy of an [[electron]] bound to an [[atom]] (at rest) is quantized. This accounts for the stability of atoms, and [[matter]] in general.


==Etymology and discovery==
An entirely new conceptual framework was developed around this idea, during the first half of the 1900s. Usually referred to as quantum "mechanics", it is regarded by virtually every professional physicst as the most fundamental framework we have for understanding and describing nature. For the very practical reason that it works. It is "in the nature of things", not a more or less arbitrary human preference.
The word {{lang|la|quantum}} is the neuter singular of the [[Latin]] interrogative adjective [[wiktionary:quantus#Latin|quantus]], meaning "how much". "{{lang|la|Quanta}}", the neuter plural, short for "quanta of electricity" (electrons), was used in a 1902 article on the [[photoelectric effect]] by [[Philipp Lenard]], who credited [[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 [[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 }} {{in lang|de}}</ref> on Mayer's work, and the word ''quantum'' can be found in the formulation of the [[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=dead |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]] and 1918 Nobel Prize for Physics recipient [[Max Planck]] (1858–1947)]]
In 1901, [[Max Planck]] used ''quanta'' to mean "quanta of matter and electricity",<ref name="Planck1901">{{cite journal |last = Planck |first = M. |author-link = Max Planck |year = 1901 |title = Ueber die Elementarquanta der Materie und der Elektricität |journal = [[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 |via=Zenodo |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 |via=Zenodo |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''), [[Albert Einstein]] suggested that [[radiation]] existed in spatially localized packets which he called [[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 = [[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 [[Max Planck]], who had been trying to understand the emission of radiation from heated objects, known as [[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 [[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 [[Planck constant]], and reported more precise values for the unit of [[electrical charge]] and the [[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.
==Discovery of quantum theory==
[[Quantum mechanics]], the branch of physics based on quantization, began in [[1900]] when [[Max Planck]] published his theory explaining the [[emission spectrum]] of [[black body|black bodies]]. In that paper, Planck, used the ''[[Planck units|Natural]]'' system of [[Physical unit|units]] invented by him the [[1898|previous year]].


==Quantization==
The consequences of the differences between [[classical mechanics|classical]] and quantum mechanics quickly became obvious. But it was not until [[1926]], by the work of [[Werner Heisenberg]], [[Erwin Schrödinger]], and others, that quantum theory became correctly formulated and understood mathematically. Despite tremendous experimental success, the philosophical interpretations of quantum theory are still widely debated.
{{main article| Quantization (physics)}}
While quantization was first discovered in [[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 -->


==See also==
Planck himself felt disturbed by the new idea of quantization, and with good reason. But, finding no alternative, he nevertheless ended up using it, and the work was well received. He called it "a few weeks of the most strenuous work of his life," eighteen years later upon receiving the [[Nobel Prize in Physics]] for his work. During those few weeks, he even had to discard much of his own theoretical work from the preceding years. Quantization turned out to be the only way to describe the new, and detailed experiments which were just then being performed. He did this practically overnight, openly reporting his change of mind to his scientific colleages, in the October, November, and December meetings of the [[German Physical Society]], in [[Berlin]], where the black body work was being intensely discussed. In this way, careful experimentalists (including [[F. Paschen]], [[O.R. Lummer]], [[E. Pringsheim]], [[H.L. Rubens]], and [[F. Kurlbaum]]), and a reluctant theorist, ushered in the greatest revolution science has ever seen.
{{cols|colwidth=16em}}
* [[Graviton]]
* [[Introduction to quantum mechanics]]
* [[Magnetic flux quantum]]
* [[Particle]]
** [[Elementary particle]]
** [[Subatomic particle]]
* [[Photon polarization]]
* [[Qubit]]
* [[Quantum cellular automata]]
* [[Quantum channel]]
* [[Quantum chromodynamics]]
* [[Quantum cognition]]
* [[Quantum coherence]]
* [[Quantum computer]]
* [[Quantum cryptography]]
* [[Quantum dot]]
* [[Quantum electronics]]
* [[Quantum entanglement]]
* [[Quantum fiction]]
* [[Quantum field theory]]
* [[Quantum lithography]]
* [[Quantum mechanics]]
* [[Quantum mind]]
* [[Quantum mysticism]]
* [[Quantum number]]
* [[Quantum optics]]
* [[Quantum sensor]]
* [[Quantum state]]
* [[Quantum suicide and immortality]]
* [[Quantum teleportation]]
{{colend}}


==References==
===The quantum black-body radiation formula===
{{Reflist}}
When a body is heated, it emits [[heat|heat radiation]], which is [[infrared radiation]], being a form of [[electromagnetic waves]]. All of this was well understood at the time, and of considerable practical importance. When the body becomes red-hot, the red [[wavelength]] parts start to become visible. This had been studied over the previous years, as the instruments were being developed. However, most of the heat radiation remains infrared, until the body becomes as hot as the surface of the [[Sun]] (about 6000 °C, where most of the light is green in color). This was not achievable in the laboratory at that time. What is more, to measure specific infrared wavelengths was only then becoming feasible, due to newly developed experimental techniques. Until then, most of the [[electromagnetic spectrum]] was not measurable, and therefore blackbody emission had not been mapped out in detail.


==Further reading==
The quantum black-body radiation formula, being the very first piece of quantum mechanics, appeared sunday evening October 7, 1900, in a so-called back-of-the-envelope calculation by Planck. It was based on a report by [[Rubens]] (visiting with his wife) of the very latest experimental findings in the infrared. Later that evening, Planck sent the formula on a postcard, which Rubens had the following morning. A couple of days later, he could tell Planck that it worked perfectly. As it does to this day.
* {{Cite book |last=Hoffmann |first=Banesh |title=The Strange story of the quantum: An account for the general reader of the growth of the ideas underlying our present atomic knowledge |date=1959 |publisher=Dover |isbn=978-0-486-20518-2 |edition=2 |location=New York}}
* {{Cite book |last=Mehra |first=Jagdish |author-link=Jagdish Mehra |title=The historical development of quantum theory. 4: Pt.1, the fundamental equations of quantum mechanics, 1925-1926 |last2=Rechenberg |first2=Helmut |author-link2=Helmut Rechenberg |last3=Mehra |first3=Jagdish |last4=Rechenberg |first4=Helmut |date=2001 |publisher=Springer |isbn=978-0-387-95178-2 |edition=1. softcover print |location=New York Heidelberg}}
* M. Planck, ''A Survey of Physical Theory'', transl. by R. Jones and D.H. Williams, Methuen & Co., Limited., London 1925 (Dover edition 17 May 2003, ISBN 978-0486678672) including the Nobel lecture.
* Rodney, Brooks (14 December 2010) ''Fields of Color: The theory that escaped Einstein''. Allegra Print & Imaging. ISBN 979-8373308427


{{Authority control}}
At first, it was just a fit to the data. Only weeks later did it turn out to enforce quantization.

That the latter became possible involved a certain amount of luck (or skill, even though Planck himself called it "a fortuitous guess at an interpolation formula"). It only had that drastic "side effect" because the formula happened to become fundamentally correct, in regard to the as yet non-existent quantum theory. And normally, that much is not at all expected. The skill lay in simplifying the mathematics, so that this could happen. And here Planck used hard won experience from the previous years. Briefly stated, he had two mathematical expressions:

*(i) from the previous work on the red parts of the spectrum, he had ''x'';
*(ii) now, from the new infrared data, he got ''x''&sup2;.

Combining these as ''x''(''a''+''x''), he still has ''x'', approximately, when ''x'' is much smaller than ''a'' ( the red end of the spectrum). But now also ''x''&sup2;, again approximately, when ''x'' is much larger than ''a'' (in the infrared). The luck part is that, this procedure turned out to actually give something completely right, far beyond what could reasonably be expected. The formula for the energy ''E'', in a single mode of radiation at frequency ''f'', and temperature ''T'', can be written

:<math>E = \frac{h f}{e^{\frac{h f}{k T}} - 1} </math>

This is (essentially) what is being compared with the experimental measurements. There are two parameters to determine from the data, written in the present form by the symbols used today: ''h'' is the new [[Planck's constant]], and ''k'' is [[Boltzmann's constant]]. Both have now become fundamental in physics, but that was by no means the case at the time. The "elementary quantum of energy" is ''hf''. But such a unit does not normally exist, and is not required for quantization.

===The birthday of quantum mechanics===
From the experiments, Planck deduced the numerical values of ''h'' and ''k''. Thus he could report, in the German Physical Society meeting on December 14, 1900, where quantization (of energy) was revealed for the first time, values of the [[Avogadro's number|Avogadro-Loschmidt number]], the number of real molecules in a [[mole (unit)|mole]], and the unit of [[electrical charge]], which were more accurate than those known until then. This event has been referred to as "the birthday of quantum mechanics".

==Quantization in antiquity==
In a sense, it can be said that the quantization idea is very old. A string under tension, and fixed at both ends, will [[oscillation|vibrate]] at certain quantized [[frequencies]], corresponding to various [[standing waves]]. This, of course, is the basis of [[music]]. The basic idea was regarded as essential by the [[Pythagoreans]], who are reported to have held numbers in high esteem.

It is a curious fact that, the famous formula, named after Pythagoras, for the side lengths of a [[right triangle]], today serves as a cornerstone of quantum mechanics as well.

The very existence of [[atoms]], [[molecules]], [[solids]], and so on, can be ascribed to various forms of quantization. Contrary to notions of matter as some form of [[continuous]] medium. This was also understood already in antiquity, particularly by [[Leucippus|Leucippos]] and [[Democritus|Democritos]], although not generally appreciated, even by physicists, really, until the invention of quantum mechanics.

It should be mentioned, though, that later works within the [[Epicurean]] school of thought played a significant role in forming the physics and chemistry of the [[Renaissance]] period in Europe. In particular the famous tutorial poem "De rerum natura" by the Roman author [[Lucretius|Titus Lucretius Carus]].

==References==
*M. Planck, ''A Survey of Physical Theory'', transl. by R. Jones and D.H. Williams, Methuen & Co., Ltd., London 1925 (Dover editions 1960 and 1993) including the Nobel lecture.
*J. Mehra and H. Rechenberg, ''The Historical Development of Quantum Theory'', Vol.1, Part 1, Springer-Verlag New York Inc., New York 1982.
*Lucretius, "On the Nature of the Universe", transl. from the Latin by R.E. Latham, Penguin Books Ltd., Harmondsworth 1951. There are, of course, many translations, and the translation's title varies. Some put emphasis on how things work, others on what things are found in nature.

==See also==
*[[Quantum mechanics]]
*[[Quantum state]]
*[[Quantum number]]
*[[Quantum cryptography]]
*[[Quantum electronics]]
*[[Quantum computing]]
*[[Magnetic flux quantum]]
*[[Quantization]]
*[[Subatomic particle]]
*[[Elementary particle]]


[[Category:Quantum mechanics]]
[[Category:Quantum mechanics]]

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Latest revision as of 13:45, 14 November 2024

In physics, a quantum (pl.: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. Quantum is a discrete quantity of energy proportional in magnitude to the frequency of the radiation it represents. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization".[1] This means that the magnitude of the physical property can take on only discrete values consisting of integer multiples of one quantum. For example, a photon is a single quantum of light of a specific frequency (or of any other form of electromagnetic radiation). Similarly, the energy of an electron bound within an atom is quantized and can exist only in certain discrete values.[2] 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 quantum mechanics. Quantization of energy and its influence on how energy and matter interact (quantum electrodynamics) is part of the fundamental framework for understanding and describing nature.

Etymology and discovery

[edit]

The word quantum is the neuter singular of the 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 photoelectric effect by Philipp Lenard, who credited Hermann von Helmholtz for using the word in the area of electricity. However, the word quantum in general was well known before 1900,[3] 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 Julius von Mayer were physicians as well as physicists. Helmholtz used quantum with reference to heat in his article[4] on Mayer's work, and the word quantum can be found in the formulation of the first law of thermodynamics by Mayer in his letter[5] dated July 24, 1841.

German physicist and 1918 Nobel Prize for Physics recipient Max Planck (1858–1947)

In 1901, Max Planck used quanta to mean "quanta of matter and electricity",[6] gas, and heat.[7] 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), Albert Einstein suggested that radiation existed in spatially localized packets which he called "quanta of light" ("Lichtquanta").[8]

The concept of quantization of radiation was discovered in 1900 by Max Planck, who had been trying to understand the emission of radiation from heated objects, known as 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"),[9] Planck accounted for certain objects changing color when heated.[10] On December 14, 1900, Planck reported his 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.[11] As a result of his experiments, Planck deduced the numerical value of h, known as the Planck constant, and reported more precise values for the unit of electrical charge and the Avogadro–Loschmidt number, the number of real molecules in a 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

[edit]

While quantization was first discovered in electromagnetic radiation, it describes a fundamental aspect of energy not just restricted to photons.[12] 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.[13]

See also

[edit]

References

[edit]
  1. ^ Wiener, N. (1966). Differential Space, Quantum Systems, and Prediction. Cambridge, Massachusetts: The Massachusetts Institute of Technology Press
  2. ^ Rovelli, Carlo (January 2017). Reality is not what it seems: the elementary structure of things. Translated by Carnell, Simon; Segre, Erica (1st American ed.). New York, New York: Riverhead Books. pp. 109–130. ISBN 978-0-7352-1392-0.
  3. ^ E. Cobham Brewer 1810–1897. Dictionary of Phrase and Fable. 1898. Archived 2017-06-30 at the Wayback Machine
  4. ^ E. Helmholtz, Robert Mayer's Priorität Archived 2015-09-29 at the Wayback Machine (in German)
  5. ^ Herrmann, Armin (1991). "Heimatseite von Robert J. Mayer" (in German). Weltreich der Physik, Gent-Verlag. Archived from the original on 1998-02-09.
  6. ^ Planck, M. (1901). "Ueber die Elementarquanta der Materie und der Elektricität". Annalen der Physik (in German). 309 (3): 564–566. Bibcode:1901AnP...309..564P. doi:10.1002/andp.19013090311. Archived from the original on 2023-06-24. Retrieved 2019-09-16 – via Zenodo.
  7. ^ Planck, Max (1883). "Ueber das thermodynamische Gleichgewicht von Gasgemengen". Annalen der Physik (in German). 255 (6): 358–378. Bibcode:1883AnP...255..358P. doi:10.1002/andp.18832550612. Archived from the original on 2021-01-21. Retrieved 2019-07-05 – via Zenodo.
  8. ^ Einstein, A. (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" (PDF). Annalen der Physik (in German). 17 (6): 132–148. Bibcode:1905AnP...322..132E. doi:10.1002/andp.19053220607. Archived (PDF) from the original on 2015-09-24. Retrieved 2010-08-26.. A partial English translation Archived 2021-01-21 at the Wayback Machine is available from Wikisource.
  9. ^ Max Planck (1901). "Ueber das Gesetz der Energieverteilung im Normalspectrum (On the Law of Distribution of Energy in the Normal Spectrum)". Annalen der Physik. 309 (3): 553. Bibcode:1901AnP...309..553P. doi:10.1002/andp.19013090310. Archived from the original on 2008-04-18.
  10. ^ Brown, T., LeMay, H., Bursten, B. (2008). Chemistry: The Central Science Upper Saddle River, New Jersey: Pearson Education ISBN 0-13-600617-5
  11. ^ Klein, Martin J. (1961). "Max Planck and the beginnings of the quantum theory". Archive for History of Exact Sciences. 1 (5): 459–479. doi:10.1007/BF00327765. S2CID 121189755.
  12. ^ Parker, Will (2005-02-11). "Real-World Quantum Effects Demonstrated". ScienceAGoGo. Retrieved 2023-08-20.
  13. ^ Modern Applied Physics-Tippens third edition; McGraw-Hill.

Further reading

[edit]
  • Hoffmann, Banesh (1959). The Strange story of the quantum: An account for the general reader of the growth of the ideas underlying our present atomic knowledge (2 ed.). New York: Dover. ISBN 978-0-486-20518-2.
  • Mehra, Jagdish; Rechenberg, Helmut; Mehra, Jagdish; Rechenberg, Helmut (2001). The historical development of quantum theory. 4: Pt.1, the fundamental equations of quantum mechanics, 1925-1926 (1. softcover print ed.). New York Heidelberg: Springer. ISBN 978-0-387-95178-2.
  • M. Planck, A Survey of Physical Theory, transl. by R. Jones and D.H. Williams, Methuen & Co., Limited., London 1925 (Dover edition 17 May 2003, ISBN 978-0486678672) including the Nobel lecture.
  • Rodney, Brooks (14 December 2010) Fields of Color: The theory that escaped Einstein. Allegra Print & Imaging. ISBN 979-8373308427