Planck–Einstein relation

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The Planck–Einstein relation,[1][2] also referred to as the Einstein relation,[1][3][4] Planck's energy–frequency relation,[5] the Planck relation,[6] and the Planck equation,[7] is a formula integral to quantum mechanics, and states that the energy of a photon (E) is proportional to its frequency (ν).

E = h \nu

The constant of proportionality, h, is known as the Planck constant. Several equivalent forms of the relation exist.

The relation accounts for quantized nature of light, and plays a key role in understand phenomena such as the photoelectric effect, and Planck's law of black body radiation. See also the Planck postulate.

Spectral forms[edit]

Light can be characterized using several spectral quantities, such as frequency (\nu), wavelength (\lambda), wavenumber (\tilde \nu) and their angular equivalents (angular frequency \omega, angular wavelength y, and angular wavenumber k). These quantities are related through

f = \frac{c}{\lambda} = c \tilde \nu = \frac{\omega}{2 \pi} = \frac{2 \pi c}{y} = \frac{ck}{2 \pi},

so the Planck relation can take the following 'standard' forms

E = h \nu = \frac{hc}{\lambda} = h c \tilde \nu,

as well as the following 'angular' forms,

E = \hbar \omega = \frac{\hbar c}{y} = \hbar c k.

The angular forms make use of the reduced Planck constant \hbar = \frac{h}{2 \pi}. Here c is the speed of light.

de Broglie relation[edit]

The de Broglie relation,[4][8][9] also known as the de Broglie's momentum–wavelength relation,[5] generalizes the Planck relation to matter waves. Louis de Broglie argued that if particles had a wave nature, the relation E = h \nu would also apply to them, and postulated that particles would have a wavelength equal to \lambda = \frac{h}{p}. Combining de Broglie's postulate with the Planck–Einstein relation leads to

p = h \tilde \nu or
p = \hbar k.

The de Broglie's relation is also often encountered in vector form

\mathbf{p} = \hbar \mathbf{k},

where \mathbf{p} is the momentum vector, and \mathbf{k} is the angular wave vector.

Bohr's frequency condition[edit]

Bohr's frequency condition states that the frequency of a photon absorbed or emitted during an electronic transition is related to the energy difference (\Delta E) between the two energy levels involved in the transition:[10]

\Delta E = h \nu.

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

References[edit]

  1. ^ a b French & Taylor (1978), pp. 24, 55.
  2. ^ Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11.
  3. ^ Messiah (1958/1961), p. 72.
  4. ^ a b Weinberg (1995), p. 3.
  5. ^ a b Schwinger (2001), p. 203.
  6. ^ Landsberg (1978), p. 199.
  7. ^ Landé (1951), p. 12.
  8. ^ Messiah (1958/1961), p. 14.
  9. ^ Cohen-Tannoudji, Diu & Laloë (1973/1977), p. 27.
  10. ^ van der Waerden (1967), p. 5.

Cited bibliography[edit]