Editing Zero-point energy (section)

=== Second quantum theory ===
[[File:Max Planck Nobel 1918.jpg|thumb|upright|Planck in 1918, the year he received the [[Nobel Prize in Physics]] for his work on [[Quantum mechanics|quantum theory]]]]
In 1900, [[Max Planck]] derived the average energy {{mvar|ε}} of a single ''energy radiator'', e.g., a vibrating atomic unit, as a function of absolute temperature:{{sfnp|Planck|1900}}
<math display="block"> \varepsilon = \frac{h\nu}{ e^{h\nu/(kT)}-1} \,,</math>
where {{mvar|h}} is the [[Planck constant]], {{mvar|ν}} is the [[frequency]], {{mvar|k}} is the [[Boltzmann constant]], and {{mvar|T}} is the absolute [[temperature]]. The zero-point energy makes no contribution to Planck's original law, as its existence was unknown to Planck in 1900.{{sfnp|Loudon|2000|p=9}}

The concept of zero-point energy was developed by [[Max Planck]] in Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900.{{sfnp|Kragh|2012|p=7}}

In 1912, Max Planck published the first journal article to describe the discontinuous emission of radiation, based on the discrete quanta of energy.{{sfnp|Planck|1912a}} In Planck's "second quantum theory" resonators absorbed energy continuously, but emitted energy in discrete energy quanta only when they reached the boundaries of finite cells in phase space, where their energies became integer multiples of {{math|''hν''}}. This theory led Planck to his new radiation law, but in this version energy resonators possessed a zero-point energy, the smallest average energy a resonator could take on. Planck's radiation equation contained a residual energy factor, one {{math|{{sfrac|''hν''|2}}}}, as an additional term dependent on the frequency {{mvar|ν}}, which was greater than zero (where {{mvar|h}} is the Planck constant). It is therefore widely agreed that "Planck's equation marked the birth of the concept of zero-point energy."{{sfnp|Milonni|1994|p=10}} In a series of papers from 1911 to 1913,<ref>See {{harvs|last1=Planck|year1=1911|year2=1912a|year3=1912b|year4=1913}} and {{harvp|Planck|1958}} for reprints</ref> Planck found the average energy of an oscillator to be:{{sfnp|Kragh|2012|p=7}}{{sfnp|Kuhn|1978|p=235}}
<math display="block">\varepsilon =\frac{h\nu} 2 + \frac{h\nu}{e^{h\nu/(kT)}-1} ~.</math>

[[File:Albert Einstein (Nobel).png|thumb|upright|Einstein's official 1921 portrait after receiving the Nobel Prize in Physics]]
Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistant [[Otto Stern]].<ref>{{cite journal|last1=Einstein|first1=Albert|last2=Stern|first2=Otto|title=Einige Argumente für die Annahme einer molekularen Agitation beim absoluten Nullpunkt|language=de|trans-title=Some arguments for the assumption of a molecular agitation at the absolute zero point|journal=Annalen der Physik|date=1913|volume=345|issue=3|pages=551–560|doi=10.1002/andp.19133450309|bibcode=1913AnP...345..551E|url=https://zenodo.org/record/1424262}}</ref> In 1913 they published a paper that attempted to prove the existence of zero-point energy by calculating the [[specific heat]] of hydrogen gas and compared it with the experimental data. However, after assuming they had succeeded, they retracted support for the idea shortly after publication because they found Planck's second theory may not apply to their example. In a letter to [[Paul Ehrenfest]] of the same year Einstein declared zero-point energy "dead as a doornail".{{sfnp|Einstein|1993|pp=563–565}} Zero-point energy was also invoked by [[Peter Debye]],<ref>{{cite journal|last1=Debye|first1=Peter|title=Interferenz von Röntgenstrahlen und Wärmebewegung|language=de|trans-title=Interference of X-rays and thermal motion|journal=Annalen der Physik|date=1913|volume=348|issue=1|pages=49–92|doi=10.1002/andp.19133480105|bibcode=1913AnP...348...49D|url=https://zenodo.org/record/1424272}}</ref> who noted that zero-point energy of the atoms of a [[crystal lattice]] would cause a reduction in the intensity of the diffracted radiation in [[X-ray diffraction]] even as the temperature approached absolute zero. In 1916 [[Walther Nernst]] proposed that empty space was filled with zero-point [[electromagnetic radiation]].<ref>{{cite journal|last1=Nernst|first1=Walther|title=Über einen Versuch, von quantentheoretischen Betrachtungen zur Annahme stetiger Energieänderungen zurückzukehren|language=de|trans-title=On an attempt to return from quantum-theoretical considerations to the assumption of constant energy changes|journal=Verhandlungen der Deutschen Physikalischen|date=1916|volume=18|pages=83–116}}</ref> With the development of general relativity Einstein found the energy density of the vacuum to contribute towards a cosmological constant in order to obtain static solutions to his field equations; the idea that empty space, or the vacuum, could have some intrinsic energy associated with it had returned, with Einstein stating in 1920:

{{blockquote|There is a weighty argument to be adduced in favour of the aether hypothesis. To deny the aether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view&nbsp;... according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether. According to the general theory of relativity space without aether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this aether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.<ref>{{cite book|last1=Einstein|first1=Albert|title=Äther und Relativitäts-Theorie|language=de|trans-title=Aether and relativity theory|date=1920|publisher=Springer|location=Berlin}}</ref><ref>{{cite book|last1=Einstein|first1=Albert|editor1-last=Jeffery|editor1-first=G. B.|editor2-last=Perrett|editor2-first=W.|title=Sidelights on Relativity: Ether and the Theory of Relativity|url=https://archive.org/details/sidelightsonrela00einsuoft|date=1922|publisher=Methuen & Co|location=New York|pages=[https://archive.org/details/sidelightsonrela00einsuoft/page/n8 1]–24}}</ref>}}

[[File:Heisenberg,Werner 1924 Göttingen - adjusted.jpeg|thumb|upright|Heisenberg, 1924]]
{{ill|Kurt Bennewitz|de|Kurt Bennewitz (Chemiker)}} and [[Francis Simon]] (1923),<ref>{{cite journal|last1=Bennewitz|first1=Kurt|last2=Simon|first2=Franz|title=Zur Frage der Nullpunktsenergie|language=de|trans-title=On the question of zero-point energy|journal=Zeitschrift für Physik|date=1923|volume=16|issue=1|doi=10.1007/BF01327389|pages=183–199|bibcode=1923ZPhy...16..183B|s2cid=121049183}}</ref> who worked at [[Walther Nernst]]'s laboratory in Berlin, studied the melting process of chemicals at low temperatures. Their calculations of the melting points of [[hydrogen]], [[argon]] and [[Mercury (element)|mercury]] led them to conclude that the results provided evidence for a zero-point energy. Moreover, they suggested correctly, as was later verified by Simon (1934),<ref>{{cite journal|last1=Simon|first1=F.|title=Behaviour of Condensed Helium near Absolute Zero|journal=Nature|date=1934|volume=133|issue=3362|page=529|doi=10.1038/133529a0|bibcode=1934Natur.133Q.529S|s2cid=4130047|doi-access=free}}</ref><ref>{{cite journal|last1=Dugdale|first1=J. S.|last2=Simon|first2=F. E.|title=Thermodynamic Properties and Melting of Solid Helium|journal=Proceedings of the Royal Society|date=1953|volume=218|issue=1134|page=291|doi=10.1098/rspa.1953.0105|bibcode=1953RSPSA.218..291D|s2cid=98061516}}</ref> that this quantity was responsible for the difficulty in solidifying helium even at absolute zero. In 1924 [[Robert S. Mulliken|Robert Mulliken]]<ref>{{cite journal|last1=Mulliken|first1=Robert S.|title=The band spectrum of boron monoxide|journal=Nature|date=1924|volume=114|issue=2862|pages=349–350|doi=10.1038/114349a0|bibcode=1924Natur.114..349M|s2cid=4121118}}</ref> provided direct evidence for the zero-point energy of molecular vibrations by comparing the band spectrum of <sup>10</sup>BO and <sup>11</sup>BO: the isotopic difference in the transition frequencies between the ground vibrational states of two different electronic levels would vanish if there were no zero-point energy, in contrast to the observed spectra. Then just a year later in 1925,<ref>{{Cite book|last1=Heisenberg|first1=W.|chapter=Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen |language=de |trans-chapter=On quantum-theoretical reinterpretation of kinematic and mechanical relationships |editor-last=Blum|editor-first=Walter|editor-last2=Rechenberg|editor-first2=Helmut|editor-link2=Helmut Rechenberg |editor-last3=Dürr|editor-first3=Hans-Peter|trans-title=Original Scientific Papers |title=Wissenschaftliche Originalarbeiten|date=1925|pages=382–396|publication-date=1985|location=Berlin, Heidelberg|publisher=Springer|doi= 10.1007/978-3-642-61659-4_26|isbn=978-3-642-64900-4|oclc= 7331244990}}</ref> with the development of [[matrix mechanics]] in [[Werner Heisenberg]]'s article "[[Quantum theoretical re-interpretation of kinematic and mechanical relations]]" the zero-point energy was derived from quantum mechanics.{{sfnp|Kragh|2002|p=162}}

In 1913 [[Niels Bohr]] had proposed what is now called the [[Bohr model]] of the atom,<ref>{{cite journal | first=Niels |last=Bohr | title=On the Constitution of Atoms and Molecules, Part I | journal=Philosophical Magazine | year=1913 | volume=26 | pages=1–24 | doi= 10.1080/14786441308634955| url=http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf | issue=151 | bibcode=1913PMag...26....1B}}</ref><ref>{{cite journal | first=Niels |last=Bohr | title=On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus | journal=Philosophical Magazine | year=1913 | volume=26 | pages=476–502 | url=http://web.ihep.su/dbserv/compas/src/bohr13b/eng.pdf | doi=10.1080/14786441308634993 | issue=153 | bibcode=1913PMag...26..476B}}</ref><ref>{{cite journal | first=Niels |last=Bohr | title=On the Constitution of Atoms and Molecules, Part III Systems containing several nuclei| journal=Philosophical Magazine | year=1913 | volume=26 | issue=155| pages=857–875 | doi=10.1080/14786441308635031| url=https://zenodo.org/record/1430922 | bibcode=1913PMag...26..857B}}</ref> but despite this it remained a mystery as to why electrons do not fall into their nuclei. According to classical ideas, the fact that an accelerating charge loses energy by radiating implied that an electron should spiral into the nucleus and that atoms should not be stable. This problem of classical mechanics was nicely summarized by [[James Hopwood Jeans]] in 1915: "There would be a very real difficulty in supposing that the (force) law {{math|{{sfrac|1|''r''<sup>2</sup>}}}} held down to the zero values of {{mvar|r}}. For the force between two charges at zero distance would be infinite; we should have charges of opposite sign continually rushing together and, when once together, no force would be adequate to separate them. [...] Thus the matter in the universe would tend to shrink into nothing or to diminish indefinitely in size."<ref>{{cite book|last1=Jeans|first1=James Hopwood|title=The mathematical theory of electricity and magnetism|url=https://archive.org/details/cu31924012330589|date=1915|publisher=Cambridge University Press|location=Cambridge|page=[https://archive.org/details/cu31924012330589/page/n179 168]|edition=3rd}}</ref> The resolution to this puzzle came in 1926 when [[Erwin Schrödinger]] introduced the [[Schrödinger equation]].<ref>{{cite journal|last1=Schrödinger|first1=Erwin|title=Quantisierung als Eigenwertproblem|language=de|trans-title=Quantization as an eigenvalue problem|journal=Annalen der Physik|date=1926|volume=79|issue=13|pages=361–376|doi=10.1002/andp.19263851302|bibcode=1926AnP...385..437S}}</ref> This equation explained the new, non-classical fact that an electron confined to be close to a nucleus would necessarily have a large kinetic energy so that the minimum total energy (kinetic plus potential) actually occurs at some positive separation rather than at zero separation; in other words, zero-point energy is essential for atomic stability.<ref>{{cite book|last1=Lieb|first1=E. H.|last2=Seiringer|first2=R.|title=The Stability of Matter in Quantum Mechanics|url=https://archive.org/details/stabilitymatterq00hlie|url-access=limited|date=2009|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-19118-0|pages=[https://archive.org/details/stabilitymatterq00hlie/page/n19 2]–3|oclc=638472161}}</ref>