Editing Interaction picture (section)

====Hamiltonian operator====

For the operator <math>H_0</math> itself, the interaction picture and Schrödinger picture coincide:
:<math>H_{0,\text{I}}(t) = \mathrm{e}^{\mathrm{i} H_{0,\text{S}} t / \hbar} H_{0,\text{S}} \mathrm{e}^{-\mathrm{i} H_{0,\text{S}} t / \hbar} = H_{0,\text{S}}.</math>
This is easily seen through the fact that operators [[commutativity|commute]] with differentiable functions of themselves. This particular operator then can be called <math>H_0</math> without ambiguity.

For the perturbation Hamiltonian <math>H_{1,\text{I}}</math>, however,
:<math>H_{1,\text{I}}(t) = \mathrm{e}^{\mathrm{i} H_{0,\text{S}} t / \hbar} H_{1,\text{S}} \mathrm{e}^{-\mathrm{i} H_{0,\text{S}} t / \hbar},</math>
where the interaction-picture perturbation Hamiltonian becomes a time-dependent Hamiltonian, unless [''H''<sub>1,S</sub>, ''H''<sub>0,S</sub>] = 0.

It is possible to obtain the interaction picture for a time-dependent Hamiltonian ''H''<sub>0,S</sub>(''t'') as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by ''H''<sub>0,S</sub>(''t''), or more explicitly with a time-ordered exponential integral.