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Mathematics for Physics

81 Planck Relation

The relation that the energy of a photon is directly proportional to its frequency, given by the product of Planck’s constant and that frequency, that is used to compute the energy of a light quantum from its frequency.

(\(E = hf\)).

Note: Also called Planck’s relation. Also called the Einstein light-quantum energy relation.

definition [d] (Planck Relation) From Knight: Einstein called each packet of energy a light quantum, and he postulated that the energy of one light quantum is directly proportional to the frequency of the light. That is, each quantum of light has energy

  • \(E = hf\) ,

where \(h\) is Planck’s constant and \(f\) is the frequency of the light.

where

  • \(E\) is the energy of one light quantum.
  • \(h\) is Planck’s constant.
  • \(f\) is the frequency of the light.

81.1 Elementary Example

81.1.1 Simple

A photon of frequency \(f = 5.0 \times 10^{14}\,\mathrm{Hz}\) has energy \(E = hf\).

\[ h = 6.626 \times 10^{-34}\,\mathrm{J\cdot s} \]

\[ E = (6.626 \times 10^{-34})(5.0 \times 10^{14}) = 3.313 \times 10^{-19}\,\mathrm{J} \]

where

  • \(E\) is the photon energy.
  • \(f\) is the frequency.

81.1.2 General

For three spectral frequencies, energy scales linearly with frequency.

\[ f_{1} = 4 \times 10^{14},\quad f_{2} = 5 \times 10^{14},\quad f_{3} = 6 \times 10^{14}\,\mathrm{Hz} \]

\[ E_{i} = h f_{i} \]

where

  • each \(E_{i}\) is the energy of a photon of frequency \(f_{i}\).