In other words, motion on the line is generated by the momentum operator.

This is a commonly encountered form of the momentum operator, though not the most general one.

For instance, the momentum operator p has the following form:

However, these terms do commute with the total angular momentum operator.

This looks very similar to the commutation relation of the position and momentum operator.

The momentum operator is an example of a differential operator.

The momentum operator is assumed to be completely classical.

In the quantum case we have the following definition for momentum operator:

In this case the generator of rotations is the angular momentum operator.

This momentum operator is in position space because the partial derivatives were taken with respect to the spatial variables.

In both cases the angular momentum operator commutes with the Hamiltonian of the system.

Then the angular momentum operator about axis is defined as:

There is no self-adjoint momentum operator for a particle moving on a half-line.

This is the translation operator reconstructed in terms of the momentum operator.

A similar formula holds for the momentum operator , in systems where it has continuous spectrum.

Examples are the kinetic energy, dipole moment, and total angular momentum operators.

These equations are used together with the energy and momentum operators, which are respectively:

There are two kinds: space-fixed and body-fixed angular momentum operators.

L is then an operator, specifically called the orbital angular momentum operator.

(There are additional restrictions as well, see angular momentum operator for details.)

The body-fixed angular momentum operators are written as .

The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy.

The momentum operator: (in the position basis).

In terms of the pseudovector angular momentum operator L:

The momentum operator in one dimension is: