The Hamiltonian acts as a generator of translation of laboratory time for short times:
The proofs in the textbooks apply, giving:
Simplifies when operators not dependent on laboratory time:
Implying:
Therefore we can use commutation relations to compute the evolution of expectation values for , E , x , and p . Use Lorentz gauge so we can interchange p and A freely.
Use:
For :
Which matches the Hamiltonian equation for coordinates if we identify:
For the space coordinates:
For energy/momenta:
For vector potential:
For canonical momenta:
Which matches Hamiltonian equation for momenta above.