Editing Matter wave (section)

=== Background ===
At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according to [[Maxwell's equations]], while matter was thought to consist of localized particles (see [[wave–particle duality#History|history of wave and particle duality]]). In 1900, this division was questioned when, investigating the theory of [[black-body radiation]], [[Max Planck]] proposed that the thermal energy of oscillating atoms is divided into discrete portions, or quanta.<ref>{{cite web|url=https://physicsworld.com/a/max-planck-the-reluctant-revolutionary/ |title=Max Planck: the reluctant revolutionary |first=Helge |last=Kragh |author-link=Helge Kragh |website=Physics World |date=2000-12-01 |access-date=2023-05-19}}</ref> Extending Planck's investigation in several ways, including its connection with the [[photoelectric effect]], [[Albert Einstein]] proposed in 1905 that light is also propagated and absorbed in quanta,<ref name="WhittakerII">{{cite book | last=Whittaker | first=Sir Edmund | title=A History of the Theories of Aether and Electricity | publisher=Courier Dover Publications | date=1989-01-01 | isbn=0-486-26126-3 | volume=2}}</ref>{{rp|87}} now called [[photon]]s. These quanta would have an energy given by the [[Planck–Einstein relation]]:
<math display="block">E = h\nu</math>
and a momentum vector <math>\mathbf{p}</math>
<math display="block">\left|\mathbf{p}\right| = p = \frac{E}{c} = \frac{h}{\lambda} ,</math>
where {{math|''ν''}} (lowercase [[Nu (letter)|Greek letter nu]]) and {{math|''λ''}} (lowercase [[Lambda|Greek letter lambda]]) denote the [[frequency]] and [[wavelength]] of the light, {{math|''c''}} the speed of light, and {{math|''h''}} the [[Planck constant]].<ref>[[Albert Einstein|Einstein, A.]] (1917). Zur Quantentheorie der Strahlung, ''Physicalische Zeitschrift'' '''18''': 121–128. Translated in 
{{cite book
 |last1=ter Haar |first1=D.
 |author-link=Dirk ter Haar
 |date=1967
 |pages=[https://archive.org/details/oldquantumtheory00haar/page/167 167–183]
 |title=The Old Quantum Theory
 |url=https://archive.org/details/oldquantumtheory00haar |url-access=registration |publisher=[[Pergamon Press]]
 |lccn=66029628
}}</ref> In the modern convention, frequency is symbolized by {{math|''f''}} as is done in the rest of this article. Einstein's postulate was verified experimentally<ref name="WhittakerII"/>{{rp|89}} by [[K. T. Compton]] and [[O. W. Richardson]]<ref name="Richardson Compton 1912 pp. 783–784">{{cite journal | last1=Richardson | first1=O. W. | last2=Compton | first2=Karl T. | title=The Photoelectric Effect | journal=Science | publisher=American Association for the Advancement of Science (AAAS) | volume=35 | issue=907 | date=1912-05-17 | issn=0036-8075 | doi=10.1126/science.35.907.783 | pages=783–784| pmid=17792421 | bibcode=1912Sci....35..783R | url=https://zenodo.org/record/1448080 }}</ref> and by A. L. Hughes<ref>Hughes, A. Ll. "XXXIII. The photo-electric effect of some compounds." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 24.141 (1912): 380–390.</ref> in 1912 then more carefully including a measurement of the [[Planck constant]] in 1916 by [[Robert Andrews Millikan|Robert Millikan]].<ref>{{cite journal
 |last1=Millikan
 |first1=R.
 |year=1916
 |title=A Direct Photoelectric Determination of Planck's "''h''"
 |journal=[[Physical Review]]
 |volume=7
 |issue=3
 |pages=355–388
 |bibcode=1916PhRv....7..355M
 |doi=10.1103/PhysRev.7.355
 |doi-access=free
 }}</ref>