Planck–Einstein relation

The Planck relation^[1][2][3] (referred to as Planck's energy–frequency relation,^[4] the Planck 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:

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

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 ${\displaystyle \hbar =h/2\pi }$. 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).

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 (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.

If ${\displaystyle \lambda \equiv {l}{l_{\text{P}}}}$, where ${\displaystyle l\in \mathbb {N} }$ and ${\displaystyle l_{\text{P}}}$ is Planck length, then

${\displaystyle E=h\nu ={\frac {2\pi }{l}}k_{\text{B}}T_{\text{P}}}$,

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant and ${\displaystyle T_{\text{P}}}$ is Planck temperature. Therefore photon energy bound is ${\displaystyle E=2\pi k_{\text{B}}T_{\text{P}}=1.229\times 10^{10}{\text{J}}}$.

de Broglie relation[]

The de Broglie relation,^[10][11][12] also known as the 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's 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 condition[]

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 (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.

References[]

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, 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.
• Flowers, P., Theopold,K., Langley, R. (n.d.). Chemistry, chapter 6, Electronic Structure and Periodic Properties of Elements, OpenStax, https://opentextbc.ca/chemistry/chapter/6-2-the-bohr-model/.