Momentum Operator - Definition (position Space)

Definition (position Space)

For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as:

where ∇ is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit.

In one spatial dimension this becomes:

This is a commonly encountered form of the momentum operator, though not the most general one. For a charged particle q in an electromagnetic field, described by the scalar potential φ and vector potential A, the momentum operator must be replaced by:

where the canonical momentum operator is the above momentum operator:

This is of course true for electrically neutral particles also, since the second term vanishes if q is zero and the original operator appears.