Heisenberg Picture - Commutator Relations

Commutator Relations

Obviously, commutator relations are quite different than in the Schrödinger picture because of the time dependency of operators. For example, consider the operators and . The time evolution of those operators depends on the Hamiltonian of the system. For the one-dimensional harmonic oscillator

The evolution of the position and momentum operators is given by:

By differentiating both equations one more time and solving them with proper initial conditions

leads to:

Now, we are ready to directly compute the commutator relations:

For, one simply gets the well-known canonical commutation relations.