Editing Interaction picture (section)

===Operators===

An operator in the interaction picture is defined as
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|equation =  <math>A_\text{I}(t) = \mathrm{e}^{\mathrm{i} H_{0,\text{S}} t / \hbar} A_\text{S}(t) \mathrm{e}^{-\mathrm{i} H_{0,\text{S}} t / \hbar}.</math>
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Note that ''A''<sub>S</sub>(''t'') will typically not depend on {{mvar|t}} and can be rewritten as just ''A''<sub>S</sub>. It only depends on  {{mvar|t}} if the operator has "explicit time dependence", for example, due to its dependence on an applied external time-varying electric field. Another instance of explicit time dependence may occur when ''A''<sub>S</sub>(''t'') is a density matrix (see below).

====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.

====Density matrix====

The [[density matrix]] can be shown to transform to the interaction picture in the same way as any other operator. In particular, let {{math|''ρ''<sub>I</sub>}} and {{math|''ρ''<sub>S</sub>}} be the density matrices in the interaction picture and the Schrödinger picture respectively. If there is probability {{math|''p<sub>n</sub>''}} to be in the physical state  |''ψ''<sub>''n''</sub>⟩, then
:<math>\begin{align}
\rho_\text{I}(t)
&= \sum_n p_n(t) \left|\psi_{n,\text{I}}(t)\right\rang \left\lang \psi_{n,\text{I}}(t)\right| \\
&= \sum_n p_n(t) \mathrm{e}^{\mathrm{i} H_{0,\text{S}} t / \hbar} \left|\psi_{n,\text{S}}(t)\right\rang \left\lang \psi_{n,\text{S}}(t)\right| \mathrm{e}^{-\mathrm{i} H_{0,\text{S}} t / \hbar} \\
&= \mathrm{e}^{\mathrm{i} H_{0,\text{S}} t / \hbar} \rho_\text{S}(t) \mathrm{e}^{-\mathrm{i} H_{0,\text{S}} t / \hbar}.
\end{align}</math>