Interaction picture

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Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
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In quantum mechanics, the interaction picture (also known as the interaction representation or Dirac picture after Paul Dirac, who introduced it)^[1][2] is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables.^[3] The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. Most field-theoretical calculations^[4] use the interaction representation because they construct the solution to the many-body Schrödinger equation as the solution to the free-particle problem plus some unknown interaction parts.

Equations that include operators acting at different times, which hold in the interaction picture, don't necessarily hold in the Schrödinger or the Heisenberg picture. This is because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others.

The interaction picture is a special case of unitary transformation applied to the Hamiltonian and state vectors.