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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: =.
The de Broglie wavelength is the wavelength, λ, associated with a particle with momentum p through the Planck constant, h: =. Wave-like behavior of matter has been experimentally demonstrated, first for electrons in 1927 and for other elementary particles , neutral atoms and molecules in the years since.
The energy and momentum of an object measured in two inertial frames in energy–momentum space – the yellow frame measures E and p while the blue frame measures E ′ and p ′. The green arrow is the four-momentum P of an object with length proportional to its rest mass m 0.
A free particle with mass in non-relativistic quantum mechanics is described by the free Schrödinger equation: (,) = (,) where ψ is the wavefunction of the particle at position r and time t . The solution for a particle with momentum p or wave vector k , at angular frequency ω or energy E , is given by a complex plane wave :
The top electron has twice the momentum, while the bottom electron has half. Note that as the momentum increases, the phase velocity decreases down to c, whereas the group velocity increases up to c, until the wave packet and its phase maxima move together near the speed of light, whereas the wavelength continues to decrease without bound. Both ...
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 ...
Diagram illustrating the relationship between the wavenumber and the other properties of harmonic waves. In the physical sciences, the wavenumber (or wave number), also known as repetency, [1] is the spatial frequency of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber).
The momentum transfer plays an important role in the evaluation of neutron, X-ray, and electron diffraction for the investigation of condensed matter. Laue-Bragg diffraction occurs on the atomic crystal lattice, conserves the wave energy and thus is called elastic scattering, where the wave numbers final and incident particles, and , respectively, are equal and just the direction changes by a ...