Interaction Picture

The interaction picture is a special time-dependent unitary frame selected by a Hamiltonian split. The general exact transformation and frame-generator term are derived in Rotating Frames. Here the Hamiltonian is split into a solvable part and an interaction:

$$H(t)=H_0+V(t).$$

The $H_0$ motion is absorbed into operators, while states evolve under the transformed interaction.

For a compact Core Formalism comparison with the Schrödinger and Heisenberg pictures, see Pictures of Motion Overview.

Why Split the Hamiltonian

The split is useful when $H_0$ is exactly solvable and $V(t)$ is the part responsible for transitions. This is the natural language of time-dependent perturbation theory, light-matter interactions, and field-theory perturbation expansions.

The split is a choice. It should be made to simplify the problem.

Definitions

Let

$$U_0(t,t_0)=e^{-iH_0(t-t_0)/\hbar}.$$

The interaction-picture state is

$$\lvert\psi_I(t)\rangle =U_0^\dagger(t,t_0)\lvert\psi_S(t)\rangle.$$

An operator becomes

$$A_I(t)=U_0^\dagger(t,t_0)A_SU_0(t,t_0).$$

Interaction Hamiltonian

The transformed interaction is

$$V_I(t) =U_0^\dagger(t,t_0)V(t)U_0(t,t_0).$$

The interaction-picture state satisfies

$$i\hbar\frac{d}{dt}\lvert\psi_I(t)\rangle =V_I(t)\lvert\psi_I(t)\rangle.$$

Thus the exact problem has been rewritten so that only the interaction drives the state.

Interaction-Picture Evolution

Define $U_I(t,t_0)$ by

$$\lvert\psi_I(t)\rangle =U_I(t,t_0)\lvert\psi_I(t_0)\rangle.$$

Then

$$U_I(t,t_0) = \mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}V_I(t')\,dt' \right].$$

Expanding this expression gives the Dyson series.

Boundary of This Page

This page defines the exact picture. Interaction Picture for Perturbation Theory applies it to Hamiltonian splitting, coefficient equations, phase estimates, degeneracy, and driven examples. Later approximation pages handle transition probabilities, Fermi’s golden rule, rotating-wave approximation, and detailed scattering calculations.

From Evolution Operators to Time-Ordered Products carries this exact picture into perturbative field theory without relocating the interaction picture’s definition.

Common Mistakes

• Thinking the interaction picture is approximate by definition.
• Forgetting to transform $V(t)$ into $V_I(t)$.
• Choosing an $H_0$ that does not simplify the problem.
• Mixing Schrödinger-picture and interaction-picture operators in the same formula.
• Dropping time ordering when $V_I(t)$ does not commute with itself at different times.

Cross-Links

• Time-Dependent Perturbation Theory and Transitions
• Interaction Picture for Perturbation Theory
• Time Ordering
• Dyson Expansion as Formal Evolution
• From Evolution Operators to Time-Ordered Products
• Time-Dependent Hamiltonians
• Schrödinger Picture
• Heisenberg Picture
• Pictures of Quantum Mechanics
• Picture Transformations
• Density Operators in Different Pictures
• Path Integrals

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.

Exercises

1. If $V(t)=0$, what happens to $\lvert\psi_I(t)\rangle$?

Solution

When $V(t)=0$, $V_I(t)=0$. The interaction-picture equation becomes

$$i\hbar\frac{d}{dt}\lvert\psi_I(t)\rangle=0.$$

So $\lvert\psi_I(t)\rangle$ is constant. All $H_0$ motion has been moved into the operators.