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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 E of a photon, known as photon energy, is proportional to its frequency ν: =.
One particle: N particles: One dimension ^ = ^ + = + ^ = = ^ + (,,) = = + (,,) where the position of particle n is x n. = + = = +. (,) = /.There is a further restriction — the solution must not grow at infinity, so that it has either a finite L 2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [1] ‖ ‖ = | |.
The classical formulae for the energy and momentum of electromagnetic radiation can be re-expressed in terms of photon events. For example, the pressure of electromagnetic radiation on an object derives from the transfer of photon momentum per unit time and unit area to that object, since pressure is force per unit area and force is the change ...
The general expression for the spin angular momentum is [1] =, where is the speed of light in free space and is the conjugate canonical momentum of the vector potential.The general expression for the orbital angular momentum of light is =, where = {,,,} denotes four indices of the spacetime and Einstein's summation convention has been applied.
The energy content of this volume element at 5 km from the station is 2.1 × 10 −10 × 0.109 = 2.3 × 10 −11 J, which amounts to 3.4 × 10 14 photons per (). Since 3.4 × 10 14 > 1, quantum effects do not play a role. The waves emitted by this station are well-described by the classical limit and quantum mechanics is not needed.
Photons with high photon energy can transform in quantum mechanics to lepton and quark pairs, the latter fragmented subsequently to jets of hadrons, i.e. protons, pions, etc.At high energies E the lifetime t of such quantum fluctuations of mass M becomes nearly macroscopic: t ≈ E/M 2; this amounts to flight lengths as large as one micrometer for electron pairs in a 100 GeV photon beam, while ...
The radiation pressure again can be seen as the transfer of each photon's momentum to the opaque surface, plus the momentum due to a (possible) recoil photon for a (partially) reflecting surface. Since an incident wave of irradiance I f over an area A has a power of I f A , this implies a flux of I f / E p photons per second per unit area ...
The energy–momentum relation is consistent with the familiar mass–energy relation in both its interpretations: E = mc 2 relates total energy E to the (total) relativistic mass m (alternatively denoted m rel or m tot), while E 0 = m 0 c 2 relates rest energy E 0 to (invariant) rest mass m 0.