For a charged particle in an electromagnetic field, described by the scalar potential and vector potential, the momentum operator must be replaced by:
32.
As it is known from classical mechanics, the momentum is the generator of translation, so the relation between translation and momentum operators is:
33.
In quantum mechanics, the "'angular momentum operator "'is one of several related operators analogous to classical angular momentum.
34.
The momentum operator must be replaced by the kinetic momentum operator, which includes a contribution from the "'A "'field:
35.
The momentum operator must be replaced by the kinetic momentum operator, which includes a contribution from the "'A "'field:
36.
In quantum mechanics, observables like kinetic energy are represented as Hamiltonian and is defined in terms of the more fundamental momentum operator \ hat p as
37.
Within a given subspace, a component of a vector operator will behave in a way proportional to the same component of the angular momentum operator.
38.
The relationship between angular momentum operators and rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics, as discussed further below.
39.
Thus, there is a unitary map from the position Hilbert space to the Segal Bargmann space that intertwines these operators with the usual position and momentum operators.
40.
Then " H " must also commute with the momentum operator, since the momentum operator can be written as a sum of scaled infinitesimal translation operators.
