{{Short description|Energy–frequency relation in quantum mechanics}} {{Use American English|date = March 2019}}

The '''Planck relation'''<ref name="FT 24">French & Taylor (1978), pp. 24, 55.</ref><ref name="CT 10">Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11.</ref><ref>{{citation |editor-last=Kalckar |editor-first=J. |title=N. Bohr: Collected Works. Volume 6: Foundations of Quantum Physics {{math|I}}, (1926–1932) |publisher=North-Holland Publ. |publication-place=Amsterdam |publication-date=1985 |volume=6 |isbn= 0 444 86712 0 |pages=7–51 |chapter=Introduction}}{{rp|39}}</ref> (referred to as '''Planck's energy–frequency relation''',<ref name="Schwinger 203">Schwinger (2001), p. 203.</ref> the '''Planck–Einstein relation''',<ref>Landsberg (1978), p. 199.</ref> '''Planck equation''',<ref>Landé (1951), p. 12.</ref> and '''Planck formula''',<ref>Griffiths, D. J. (1995), pp. 143, 216.</ref> though the latter might also refer to Planck's law<ref>Griffiths, D. J. (1995), pp. 217, 312.</ref><ref>Weinberg (2013), pp. 24, 28, 31.</ref>) is a fundamental equation in quantum mechanics which states that the photon energy {{mvar|E}} is proportional to the photon frequency {{mvar|ν}} (or {{mvar|f}}): <math display="block">E = h \nu = h f.</math> The constant of proportionality, {{math|''h''}}, is known as the Planck constant. Several equivalent forms of the relation exist, including in terms of angular frequency {{mvar|ω}}: <math display="block">E = \hbar \omega,</math> where the reduced Planck constant is <math>\hbar = h / 2 \pi</math>.

The relation accounts for the quantized nature of light and plays a key role in understanding phenomena such as the photoelectric effect and black-body radiation (where the related Planck postulate can be used to derive Planck's law).

==Spectral forms== Light can be characterized using several spectral quantities, such as frequency {{mvar|ν}}, wavelength {{mvar|λ}}, wavenumber <math>\tilde{\nu}</math>, and their angular equivalents (angular frequency {{mvar|ω}}, angular wavelength {{mvar|y}}, and angular wavenumber {{mvar|k}}). These quantities are related through <math display="block">\nu = \frac{c}{\lambda} = c \tilde \nu = \frac{\omega}{2 \pi} = \frac{c}{2 \pi y} = \frac{ck}{2 \pi},</math> so the Planck relation can take the following "standard" forms: <math display="block">E = h \nu = \frac{hc}{\lambda} = h c \tilde \nu,</math> as well as the following "angular" forms: <math display="block">E = \hbar \omega = \frac{\hbar c}{y} = \hbar c k.</math>

The standard forms make use of the Planck constant {{mvar|h}}. The angular forms make use of the reduced Planck constant {{math|1=''ħ'' = {{sfrac|''h''|2π}}}}. Here {{mvar|c}} is the speed of light.

==de Broglie relation== {{See also|Matter wave#de Broglie relations}} The de Broglie relation,<ref name="Weinberg 3">Weinberg (1995), p. 3.</ref><ref>Messiah (1958/1961), p. 14.</ref><ref>Cohen-Tannoudji, Diu & Laloë (1973/1977), p. 27.</ref> also known as de Broglie's momentum–wavelength relation,<ref name="Schwinger 203"/> generalizes the Planck relation to matter waves. Louis de Broglie argued that if particles had a wave nature, the relation {{math|1=''E'' = ''hν''}} would also apply to them, and postulated that particles would have a wavelength equal to {{math|1=''λ'' = {{sfrac|''h''|''p''}}}}. Combining de Broglie's postulate with the Planck–Einstein relation leads to <math display="block">p = h \tilde \nu</math> or <math display="block">p = \hbar k.</math>

The de Broglie relation is also often encountered in vector form <math display="block">\mathbf{p} = \hbar \mathbf{k},</math> where {{math|'''p'''}} is the momentum vector, and {{math|'''k'''}} is the angular wave vector.

==Bohr's frequency condition== Bohr's frequency condition<ref>Flowers et al. (n.d), 6.2 The Bohr Model</ref> states that the frequency of a photon absorbed or emitted during an electronic transition is related to the energy difference ({{math|Δ''E''}}) between the two energy levels involved in the transition:<ref>van der Waerden (1967), p. 5.</ref> <math display="block">\Delta E = h \nu. </math>

This is a direct consequence of the Planck–Einstein relation.

==See also== * Compton wavelength

==References== {{reflist|3}}

==Cited bibliography== *Cohen-Tannoudji, C., Diu, B., Laloë, F. (1973/1977). ''Quantum Mechanics'', translated from the French by S.R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, {{ISBN|0471164321}}. *French, A.P., Taylor, E.F. (1978). ''An Introduction to Quantum Physics'', Van Nostrand Reinhold, London, {{ISBN|0-442-30770-5}}. *Griffiths, D.J. (1995). ''Introduction to Quantum Mechanics'', Prentice Hall, Upper Saddle River NJ, {{ISBN|0-13-124405-1}}. *Landé, A. (1951). ''Quantum Mechanics'', Sir Isaac Pitman & Sons, London. *Landsberg, P.T. (1978). ''Thermodynamics and Statistical Mechanics'', Oxford University Press, Oxford UK, {{ISBN|0-19-851142-6}}. *Messiah, A. (1958/1961). [https://archive.org/details/QuantumMechanicsVolumeI ''Quantum Mechanics''], volume 1, translated from the French by G.M. Temmer, North-Holland, Amsterdam. *Schwinger, J. (2001). ''Quantum Mechanics: Symbolism of Atomic Measurements'', edited by B.-G. Englert, Springer, Berlin, {{ISBN|3-540-41408-8}}. *van der Waerden, B.L. (1967). ''Sources of Quantum Mechanics'', edited with a historical introduction by B.L. van der Waerden, North-Holland Publishing, Amsterdam. *Weinberg, S. (1995). ''The Quantum Theory of Fields'', volume 1, ''Foundations'', Cambridge University Press, Cambridge UK, {{ISBN|978-0-521-55001-7}}. *Weinberg, S. (2013). ''Lectures on Quantum Mechanics'', Cambridge University Press, Cambridge UK, {{ISBN|978-1-107-02872-2}}.

{{Albert Einstein}} {{DEFAULTSORT:Planck-Einstein relation}} Category:Foundational quantum physics Category:Max Planck Category:Old quantum theory