Planck relation

The Planck relation¹²³ (referred to as Planck's energy–frequency relation,⁴ the Planck–Einstein relation,⁵ Planck equation,⁶ and Planck formula,⁷ though the latter might also refer to Planck's law⁸⁹) is a fundamental equation in quantum mechanics which states that the photon energy E is proportional to the photon frequency ν (or f): $${\displaystyle E=h\nu =hf.}$$ The constant of proportionality, h, is known as the Planck constant. Several equivalent forms of the relation exist, including in terms of angular frequency ω: $${\displaystyle E=\hbar \omega ,}$$ where the reduced Planck constant is ${\displaystyle \hbar =h/2\pi }$.

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).

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

The standard forms make use of the Planck constant h. The angular forms make use of the reduced Planck constant ħ = ⁠h/2π⁠. Here c is the speed of light.

The de Broglie relation,¹⁰¹¹¹² also known as de Broglie's momentum–wavelength relation,⁴ generalizes the Planck relation to matter waves. Louis de Broglie argued that if particles had a wave nature, the relation E = hν would also apply to them, and postulated that particles would have a wavelength equal to λ = ⁠h/p⁠. Combining de Broglie's postulate with the Planck–Einstein relation leads to $${\displaystyle p=h{\tilde {\nu }}}$$ or $${\displaystyle p=\hbar k.}$$

The de Broglie relation is also often encountered in vector form $${\displaystyle \mathbf {p} =\hbar \mathbf {k} ,}$$ where p is the momentum vector, and k is the angular wave vector.

Bohr's frequency condition¹³ states that the frequency of a photon absorbed or emitted during an electronic transition is related to the energy difference (ΔE) between the two energy levels involved in the transition:¹⁴ $${\displaystyle \Delta E=h\nu .}$$

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

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