Quantum mechanical

In quantum mechanics, momentum is defined as a self-adjoint operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables.

For a single particle described in the position basis the momentum operator can be written as

${\displaystyle \mathbf{p}=\frac{\mathrm{\hslash <!--\; \hslash \; -->}}{i}\mathrm{\nabla <!--\; \nabla \; -->}=-<!--\; -\; -->i\mathrm{\hslash <!--\; \hslash \; -->}\mathrm{\nabla <!--\; \nabla \; -->},}$

where ∇ is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in momentum space the momentum operator is represented as

${\displaystyle \mathbf{p}\mathrm{\psi <!--\; \psi \; -->}(p)=p\mathrm{\psi <!--\; \psi \; -->}(p),}$

where the operator p acting on a wave function ψ(p) yields that wave function multiplied by the value p, in an analogous fashion to the way that the position operator acting on a wave function ψ(x) yields that wave function multiplied by the value x.

For both massive and massless objects, relativistic momentum is related to the phase constant ${\displaystyle \mathrm{\beta <!--\; \beta \; -->}}$ by

${\displaystyle p=\mathrm{\hslash <!--\; \hslash \; -->}\mathrm{\beta <!--\; \beta \; -->}}$

Electromagnetic radiation (including visible light, ultraviolet light, and radio waves) is carried by photons. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the solar sail. The calculation of the momentum of light within dielectric media is somewhat controversial (see Abraham–Minkowski controversy).