Derivation of De brogle wavelenth

de Broglie relations

Quantum mechanics

The de Broglie equations relate the wavelength λ to the momentum p, and frequency f to the kinetic energy E (excluding its rest energy and any potential energy) of a particle:^[2]

where h is Planck's constant. The equation can be equivalently written as

using the definitions

• is the reduced Planck's constant (also known as Dirac's constant, pronounced "h-bar"),
• is the angular wavenumber,
• is the angular frequency.

In each pair, the second is also referred to as the Planck-Einstein relation, since it was also proposed by Planck and Einstein. E

Special relativity

Using the relativistic momentum formula from special relativity

allows the equations to be written as^[4]

where m₀ is the particle's rest mass, v is the particle's velocity, γ is the Lorentz factor, and c is the speed of light in a vacuum. See group velocity for details of the derivation of the de Broglie relations. Group velocity (equal to the particle's speed) should not be confused with phase velocity (equal to the product of the particle's frequency and its wavelength). In the case of a non-dispersive medium, they happen to be equal, but otherwise they are not.

Four-vectors

Main article: Four-vector

Using the four-momentum P = (E/c, p) and the four-wavevector K = (ω/c, k), the De Broglie relations form a single equation:

which is frame-independent