Editing Matter wave (section)

=== De Broglie hypothesis ===
[[File:Propagation of a de broglie wave.svg|290px|right|thumb|Propagation of '''de Broglie waves''' in one dimension – real part of the [[complex number|complex]] amplitude is blue, imaginary part is green. The probability (shown as the color [[opacity (optics)|opacity]]) of finding the particle at a given point {{math|''x''}} is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the [[slope]] decreases, so the amplitude diminishes again, and vice versa. The result is an alternating amplitude: a wave. Top: [[plane wave]]. Bottom: [[wave packet]].]]

{{blockquote|When I conceived the first basic ideas of wave mechanics in 1923–1924, I was guided by the aim to perform a real physical synthesis, valid for all particles, of the coexistence of the wave and of the corpuscular aspects that Einstein had introduced for photons in his theory of light quanta in 1905.|de Broglie<ref>{{cite journal |first=Louis |last=de Broglie |title=The reinterpretation of wave mechanics |journal=Foundations of Physics |volume=1 |pages=5–15 |number=1 |year=1970|doi=10.1007/BF00708650 |bibcode=1970FoPh....1....5D |s2cid=122931010 }}</ref>}}

[[Louis de Broglie|De Broglie]], in his 1924 PhD thesis,<ref name=Broglie>{{cite web |last1=de Broglie |first1=Louis Victor |title=On the Theory of Quanta |url=https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf |access-date=25 February 2023 |website=Foundation of Louis de Broglie |edition=English translation by A.F. Kracklauer, 2004.}}</ref> proposed that just as light has both wave-like and particle-like properties, [[electron]]s also have wave-like properties.
His thesis started from the hypothesis, "that to each portion of energy with a [[Invariant mass|proper mass]] {{math|''m''<sub>0</sub>}} one may associate a periodic phenomenon of the frequency {{math|''ν''<sub>0</sub>}}, such that one finds: {{math|1=''hν''<sub>0</sub> = ''m''<sub>0</sub>''c''<sup>2</sup>}}. The frequency {{math|''ν''<sub>0</sub>}} is to be measured, of course, in the rest frame of the energy packet. This hypothesis is the basis of our theory."<ref>{{cite journal | last1 = de Broglie | first1 = L. | author-link = Louis de Broglie | year = 1923 | title = Waves and quanta | journal = Nature | volume = 112 | issue = 2815| page = 540 | doi=10.1038/112540a0| bibcode = 1923Natur.112..540D| s2cid = 4082518 | doi-access = free }}</ref><ref name=Broglie />{{rp|p=8}}<ref name="Medicus">{{cite journal | last1 = Medicus | first1 = H.A. | year = 1974 | title = Fifty years of matter waves | journal = Physics Today | volume = 27 | issue = 2| pages = 38–45 | doi=10.1063/1.3128444| bibcode = 1974PhT....27b..38M}}</ref><ref name="MacKinnon">[http://scitation.aip.org/content/aapt/journal/ajp/44/11/10.1119/1.10583 MacKinnon, E. (1976). De Broglie's thesis: a critical retrospective, ''Am. J. Phys.'' '''44''': 1047–1055].</ref><ref>{{cite journal | last1 = Espinosa | first1 = J.M. | year = 1982 | title = Physical properties of de Broglie's phase waves | journal = Am. J. Phys. | volume = 50 | issue = 4| pages = 357–362 | doi=10.1119/1.12844| bibcode = 1982AmJPh..50..357E}}</ref><ref>{{cite journal | last1 = Brown | first1 = H.R. | last2 = Martins | year = 1984 | title = De Broglie's relativistic phase waves and wave groups | url = http://repositorio.unicamp.br/jspui/handle/REPOSIP/79307 | journal = Am. J. Phys. | volume = 52 | issue = 12 | pages = 1130–1140 | doi = 10.1119/1.13743 | bibcode = 1984AmJPh..52.1130B | access-date = 16 December 2019 | archive-date = 29 July 2020 | archive-url = https://web.archive.org/web/20200729040701/http://repositorio.unicamp.br/jspui/handle/REPOSIP/79307 | url-access = subscription }}</ref> (This frequency is also known as [[Compton wavelength|Compton frequency]].)

To find the [[wavelength]] equivalent to a moving body, de Broglie<ref name="WhittakerII"/>{{rp|214}} set the [[Energy–momentum relation#Connection to E = mc2|total energy]] from [[special relativity]] for that body equal to {{math | ''h&nu;''}}:
<math display="block">E = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} = h\nu</math>

(Modern physics no longer uses this form of the total energy; the [[energy–momentum relation]] has proven more useful.) De Broglie identified the velocity of the particle, {{math|''v''}}, with the wave [[group velocity]] in free space:
<math display="block"> v_\text{g} \equiv \frac{\partial \omega}{\partial k} = \frac{d\nu}{d(1/\lambda)} </math>

(The modern definition of group velocity uses angular frequency {{mvar|ω}} and wave number {{mvar|k}}). By applying the differentials to the energy equation and identifying the [[Momentum#Relativistic|relativistic momentum]]:
<math display="block"> p = \frac{mv}{\sqrt{1-\frac{v^2}{c^2}}} </math>

then integrating, de Broglie arrived at his formula for the relationship between the [[wavelength]], {{mvar|λ}}, associated with an electron and the modulus of its [[momentum]], {{math|''p''}}, through the [[Planck constant]], {{math|''h''}}:<ref>{{cite book |title=Introducing Quantum Theory |author1=McEvoy, J. P. |author2=Zarate, Oscar  |publisher=Totem Books |year=2004 |isbn=978-1-84046-577-8 |pages=110–114}}</ref>
<math display="block"> \lambda = \frac{h}{p}.</math>