Interaction picture - Definition and Overview

In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate 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.

Switching pictures

To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts, H = H₀ + H₁. Then the state vector is defined as:

<math> | \psi_{I}(t) \rang = e^{i H_{0} t / \hbar} | \psi_{S}(t) \rang <math>

Operators transform between the pictures as

<math>A_{I} = e^{i H_{0} t / \hbar} A_{S} e^{-i H_{0} t / \hbar}<math>.

The Schrödinger equation then becomes in this picture:

<math> i \hbar \frac{d}{dt} | \psi_{I} (t) \rang = H_{1, I} | \psi_{I} (t) \rang <math>

The purpose of this picture is to shunt all the time dependence due to H₀ onto the operators, leaving only H_{1, I} affecting the time-dependence of the state vectors.

The interaction picture is convenient when considering the effect of a small interaction term, H₁, being added to the Hamiltonian of a solved system, H₀. By switching into the interaction picture, you can use time-dependent perturbation theory to find the effect of H_{1, I}.

References

• Townsend, John S. (2000). A Modern Approach to Quantum Mechanics (2nd ed.). Sausalito, CA: University Science Books. ISBN 1-891389-13-0.

See also

• Bra-ket notation
• Schrödinger equation
• Haag's theorem