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

, ω:

$${\displaystyle E=\hbar \omega }$$

where ${\displaystyle \hbar =h/2\pi }$. Written using the symbol f for frequency, the relation is as follows:

$${\displaystyle E=hf}$$

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 ν, wavelength λ, wavenumber ${\displaystyle \scriptstyle {\tilde {\nu }}}$, and their angular equivalents (

k). These quantities are related through so the Planck relation can take the following 'standard' forms as well as the following 'angular' forms,

The standard forms make use of the

.

de Broglie relation

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

or

The de Broglie relation is also often encountered in

form where p is the momentum vector, and k is the .

Bohr's frequency condition

Bohr's frequency condition

$${\displaystyle \Delta E=h\nu .}$$

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

See also

• Compton wavelength

References

Cited bibliography

• ISBN 0471164321
  .
• 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.
• 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.
• ISBN 978-0-521-55001-7
  .
• ISBN 978-1-107-02872-2
  .