Momentum operator

Summary

The momentum operator is the observable that corresponds to the momentum of a particle. On $L^2(\R)$ it acts on smooth wave functions by $P\psi = -i\hbar\, \psi'$. The Fourier transform turns it into multiplication by $\hbar k$, so it is self-adjoint with spectrum $\R$, like the position operator. Position and momentum satisfy the canonical commutation relation, which leads to the uncertainty principle.