The Interaction Picture

The interaction, or Dirac, picture is a hybrid between the Schrödinger and Heisenberg pictures. In this picture, both the operators and the state vectors are time dependent; the time dependence is split between the vectors and the operators. This is achieved by splitting the Hamiltonian $\hat{H}$ into two parts - an exactly solvable, or "bare", part $\hat{H}_0$ and a "peturbation", $\hat{V}(t):$

$\hat{H}=\hat{H}_0+\hat{V}(t)$

Let us now take a solution $|\Psi(t)\rangle$ of the Schrödinger equation for the full Hamiltonian and "factor out" the time dependence due to the "bare" part of the Hamiltonian:

$|\Psi(t)\rangle=e^{-i\hat{H}_0t/\hbar}|\Psi_I(t)\rangle$

In this way, we have defined the state vector $|\Psi_I(t)\rangle$ in the interaction picture. If we substitute this into the Schrödinger equation, we find that this vector satisfies

$i\hbar\frac{d}{dt}|\Psi_I(t)\rangle=\hat{V}_I(t)|\Psi_I(t)\rangle,$

where $\hat{V}_I(t)=e^{i\hat{H}_0t/\hbar}\hat{V}(t)e^{-i\hat{H}_0t/\hbar}$ is the "perturbation" in the interaction picture. In other words, the time evolution of the state vector in the interaction picture is governed entirely by the "perturbation" part of the Hamiltonian.

We may see that the same relation between the "perturbation" in the interaction picture and the same in the Schrödinger picture is also satisfied by all operators. If we consider the expectation value of an operator, we may rewrite it in terms of the interaction picture state vectors as follows:

$\langle\hat{A}\rangle(t)=\langle\Psi(t)|\hat{A}|\Psi(t)\rangle=\langle\Psi_I(t)|e^{i\hat{H}_0t/\hbar}\hat{A}e^{-i\hat{H}_0t/\hbar}|\Psi_I(t)\rangle$

Similarly to how we defined the Heisenberg picture operators, we may define the operator $\hat{A}_I(t)$ in the interaction picture as

$\hat{A}_I(t)=e^{i\hat{H}_0t/\hbar}\hat{A}e^{-i\hat{H}_0t/\hbar}.$

We therefore see that the time dependence of operators in the interaction picture is dictated entirely by the "bare" part of the Hamiltonian.

We may derive equations of motion for operators in the interaction picture, just as we did for the Heisenberg picture; these equations are

$\frac{d}{dt}\hat{A}_{I}(t)=-\frac{i}{\hbar}\left [\hat{A},\hat{H}_0\right ]+\left (\frac{\partial\hat{A}}{\partial t}\right )_{I}.$