Momentum operator ↩

Metadata

shorthands: {}
aliases: [Canonical momentum operator]
created: 2021-12-11 07:34:31
modified: 2022-01-10 04:13:03

In quantum mechanics, we define the observable momentum operator

on the Hilbert space of wave functions as:

Where is the nabla operator and is the reduced Planck constant.

1 dimensional example

In 1D, for a plane wave with wavenumber , the momentum is:

Which satisfies the de Broglie hypothesis:

A word about square integrability of plane waves

Now we can see that is not part of the space , so we cannot use the usual inner product on it, but we can express any wave function as a superposition of plane waves. If we apply the unitary Fourier transform on the wave function, we get a that is in now, so we can take the expectation value of .