# Planck relation

The **Planck relation**^{[1]}^{[2]}^{[3]} (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 law^{[8]}^{[9]}) is a fundamental equation in quantum mechanics which states that the energy of a photon, *E*, known as photon energy, is proportional to its frequency, *ν*:

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*:

where . 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[edit]

Light can be characterized using several spectral quantities, such as frequency *ν*, wavelength *λ*, wavenumber , and their angular equivalents (angular frequency *ω*, angular wavelength *y*, and angular wavenumber *k*). These quantities are related through

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.

## de Broglie relation[edit]

The de Broglie relation,^{[10]}^{[11]}^{[12]} 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

The de Broglie relation is also often encountered in vector form

**p**is the momentum vector, and

**k**is the angular wave vector.

## Bohr's frequency condition[edit]

Bohr's frequency condition^{[13]} states that the frequency of a photon absorbed or emitted during an electronic transition is related to the energy difference (Δ*E*) between the two energy levels involved in the transition:^{[14]}

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

## See also[edit]

## References[edit]

**^**French & Taylor (1978), pp. 24, 55.**^**Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11.**^**Kalckar 1985 , p. 39.- ^
^{a}^{b}Schwinger (2001), p. 203. **^**Landsberg (1978), p. 199.**^**Landé (1951), p. 12.**^**Griffiths, D.J. (1995), pp. 143, 216.**^**Griffiths, D.J. (1995), pp. 217, 312.**^**Weinberg (2013), pp. 24, 28, 31.**^**Weinberg (1995), p. 3.**^**Messiah (1958/1961), p. 14.**^**Cohen-Tannoudji, Diu & Laloë (1973/1977), p. 27.**^**Flowers et al. (n.d), 6.2 The Bohr Model**^**van der Waerden (1967), p. 5.

## Cited bibliography[edit]

- 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).
*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.