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Codex  0 Created 2026-09-28 Updated 2026-11-05

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}

U_{0} (t, t_{0}) = e^{- i H_{0} (t - t_{0}) /ℏ}, ∣ ψ_{I} (t)⟩ = U_{0} (t, t_{0})^{†} ∣ ψ_{S} (t)⟩,

where

ℏ

S

A_{I} (t) = U_{0}^{†} A_{S} (t) U_{0}

. Differentiating the transformed quantum state and using the Schrödinger equation cancels the

H_{0}

terms and gives

i ℏ \frac{d}{d t} ∣ ψ_{I} (t)⟩ = V_{I} (t) ∣ ψ_{I} (t)⟩, V_{I} (t) = U_{0}^{†} V (t) U_{0} .

U_{I} (t, t_{0}) = T exp [- \frac{i}{ℏ} \int_{t_{0}}^{t} V_{I} (s) d s] .

The Dyson series expands this time-ordered expression in powers of the interaction. ^[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:

⟨ ψ_{S} ∣ A_{S} ∣ ψ_{S} ⟩ = ⟨ ψ_{I} ∣ A_{I} ∣ ψ_{I} ⟩

. For time-dependent

H_{0} (t)

, use its exact unitary time evolution in place of the simple matrix exponential.

Interaction picture