= Interaction picture
{wiki}
The <interaction picture> divides <quantum dynamics> between an exactly handled reference <Hamiltonian> and an interaction. Write $H(t)=H_0+V(t)$ with time-independent $H_0$, and define
$$
U_0(t,t_0)=e^{-iH_0(t-t_0)/\hbar},\qquad
|\psi_I(t)\rangle=U_0(t,t_0)^\dagger|\psi_S(t)\rangle,
$$
where $\hbar$ is the <reduced Planck constant> and the subscript $S$ denotes the <Schrödinger picture>. A <linear operator> transforms as $A_I(t)=U_0^\dagger A_S(t)U_0$. Differentiating the transformed <quantum state> and using the <Schrödinger equation> cancels the $H_0$ terms and gives
$$
i\hbar\frac{d}{dt}|\psi_I(t)\rangle=V_I(t)|\psi_I(t)\rangle,\qquad
V_I(t)=U_0^\dagger V(t)U_0.
$$
The resulting <unitary time evolution> is
$$
U_I(t,t_0)=\mathcal T\exp\left[-\frac{i}{\hbar}\int_{t_0}^tV_I(s)\,ds\right].
$$
The <Dyson series> expands this time-ordered expression in powers of the interaction. <Lectures on the Quantum Hall Effect by David Tong>[Section 2.2.3, printed pp. 54–55, equation 2.10; reference evolution and interaction-picture state equation]{ref}
This is useful when $H_0$ has a known <eigenbasis> of <energy eigenstates> and the interaction is weak enough for <perturbation theory>. The transformation itself is exact and preserves <expectation values>: $\langle\psi_S|A_S|\psi_S\rangle=\langle\psi_I|A_I|\psi_I\rangle$. For time-dependent $H_0(t)$, use its exact <unitary time evolution> in place of the simple <matrix exponential>.
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