Then " H " must also commute with the momentum operator, since the momentum operator can be written as a sum of scaled infinitesimal translation operators.
42.
The proof is trivial, all you have to do is observe that the total momentum operator is the generator of translations and that the Hamiltonian is invariant under translations.
43.
In both cases the angular momentum operator uncertainty relation this means that the angular momentum and the energy ( eigenvalue of the Hamiltonian ) can be measured at the same time.
44.
The reason for this is that the eigenstates ( or state of the system corresponding to an observed value ) of the position operator are not eigenstates of the momentum operator.
45.
Specifically, under the assumption that the classical position and momentum operators commute, the Liouville equation for the KvN wavefunction is recovered from averaged Newton's laws of motion.
46.
Where " ? " is the Minkowski metric tensor . ( It is common to drop any hats for the four-momentum operators in the commutation relations ).
47.
One can then ask whether this sinusoidal oscillation should be reflected in the state vector | \ psi \ rangle, the momentum operator \ hat { p }, or both.
48.
Replacing the energy by the energy operator and the momentum by the momentum operator in the four-momentum, one obtains the four-momentum operator, used in relativistic wave equations.
49.
Replacing the energy by the energy operator and the momentum by the momentum operator in the four-momentum, one obtains the four-momentum operator, used in relativistic wave equations.
50.
We assume that the states of the subsystems can be chosen as eigenstates of their angular momentum operators ( and of their component along any arbitrary " z " axis ).
