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.
Read more about this topic: Heisenberg Picture
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