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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“Children, who play life, discern its true law and relations more clearly than men, who fail to live it worthily, but who think that they are wiser by experience, that is, by failure.”
—Henry David Thoreau (1817–1862)