Momentum operator

The momentum operator generates spatial translations. In a one-dimensional position representation in quantum mechanics, it acts asApplying it to a plane wave $e^{ikx}$ gives momentum eigenvalue $\hslash k$, connecting momentum with the de Broglie wavelength. A localized wave packet generally contains several such momenta.

A differential expression alone does not define a self-adjoint operator: its domain and boundary conditions matter. On a periodic interval, the allowed momentum eigenvalues are discrete; on the whole real line they form a continuous spectrum. In a magnetic field, this operator is the canonical momentum. The mechanical momentum entering the kinetic energy is instead $P-qA$ in SI units, where $A$ is the vector potential.

One dimension in position representation:

In three dimensions In position representation, we define it by using the gradient, and so we see that