Planck relation

The Planck relation (referred to as Planck's energy–frequency relation, the Planck–Einstein relation, Planck equation, and Planck formula, though the latter might also refer to Planck's law) is a fundamental equation in quantum mechanics which states that the photon energy Template:Mvar is proportional to the photon frequency Template:Mvar (or Template:Mvar): $$E=h\nu =hf.$$ The constant of proportionality, Template:Math, is known as the Planck constant. Several equivalent forms of the relation exist, including in terms of angular frequency Template:Mvar: $$E=\hslash \omega ,$$ where the reduced Planck constant is $\hslash =h/2\pi$.

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).

Spectral forms

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

The standard forms make use of the Planck constant Template:Mvar. The angular forms make use of the reduced Planck constant Template:Math. Here Template:Mvar is the speed of light.

de Broglie relation

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

The de Broglie relation is also often encountered in vector form $$𝐩=\hslash 𝐤,$$ where Template:Math is the momentum vector, and Template:Math is the angular wave vector.

Bohr's frequency condition

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

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

See also

• Compton wavelength

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, Template:ISBN.
• French, A.P., Taylor, E.F. (1978). An Introduction to Quantum Physics, Van Nostrand Reinhold, London, Template:ISBN.
• Griffiths, D.J. (1995). Introduction to Quantum Mechanics, Prentice Hall, Upper Saddle River NJ, Template:ISBN.
• Landé, A. (1951). Quantum Mechanics, Sir Isaac Pitman & Sons, London.
• Landsberg, P.T. (1978). Thermodynamics and Statistical Mechanics, Oxford University Press, Oxford UK, Template:ISBN.
• 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, Template:ISBN.
• 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, Template:ISBN.
• Weinberg, S. (2013). Lectures on Quantum Mechanics, Cambridge University Press, Cambridge UK, Template:ISBN.