The Interaction Picture

The interaction picture (also called the Dirac picture) is the natural language for time-dependent perturbation theory. It sits between the two pictures you already know and removes the "boring" free evolution so that the only motion left is the one caused by the perturbation.

Recalling the Schrödinger and Heisenberg pictures

In the Schrödinger picture states carry all the time dependence and operators are fixed; in the Heisenberg picture operators evolve and states are fixed. The interaction picture splits the difference, attaching the known part of the evolution to the operators and leaving the perturbation-driven part on the states.

Start from $\hat H = \hat H_0 + \hat H'(t)$ and define the unitary generated by the unperturbed Hamiltonian, $\hat U_0(t) = e^{-i\hat H_0 t/\hbar}$ . The interaction-picture state is obtained by "undoing" this free evolution:

|\Psi_I(t)\rangle \equiv \hat U_0^\dagger(t)\,|\Psi_S(t)\rangle = e^{+i\hat H_0 t/\hbar}\,|\Psi_S(t)\rangle ,

where $|\Psi_S(t)\rangle$ is the ordinary Schrödinger-picture state.

Operators in the interaction picture

To keep matrix elements of any observable unchanged, operators are transformed the opposite way:

\hat A_I(t) \equiv \hat U_0^\dagger(t)\,\hat A_S\,\hat U_0(t) = e^{+i\hat H_0 t/\hbar}\,\hat A_S\,e^{-i\hat H_0 t/\hbar}.

In particular the perturbation becomes

\hat H'_I(t) = e^{+i\hat H_0 t/\hbar}\,\hat H'(t)\,e^{-i\hat H_0 t/\hbar}.

Expectation values are picture-independent, as they must be: $\langle \Psi_I | \hat A_I | \Psi_I\rangle = \langle \Psi_S | \hat A_S | \Psi_S\rangle$ .

The equation of motion: only the perturbation drives the state

Differentiate $|\Psi_I(t)\rangle = e^{+i\hat H_0 t/\hbar}|\Psi_S(t)\rangle$ and use the Schrödinger equation $i\hbar\,\partial_t|\Psi_S\rangle = (\hat H_0 + \hat H')|\Psi_S\rangle$ :

i\hbar\,\frac{\partial}{\partial t}\,|\Psi_I(t)\rangle = \hat H'_I(t)\,|\Psi_I(t)\rangle .

This is the central result. The free Hamiltonian $\hat H_0$ has completely dropped out of the state's equation of motion — it now lives only inside $\hat H'_I(t)$ . If the perturbation vanishes, $|\Psi_I\rangle$ is constant: the interaction picture freezes the trivial stationary-state phases, leaving behind only the transitions we care about.

Matrix elements and Bohr frequencies

Projecting the equation of motion onto an eigenstate $|f\rangle$ of $\hat H_0$ and inserting a complete set $\sum_i |i\rangle\langle i|$ gives coupled equations for the coefficients:

i\hbar\,\dot c_f(t) = \sum_i \langle f|\hat H'(t)|i\rangle\, e^{\,i\omega_{fi} t}\, c_i(t), \qquad \omega_{fi} \equiv \frac{E_f - E_i}{\hbar}.

The quantities $\omega_{fi}$ are the Bohr (transition) frequencies of the unperturbed system; they come directly from the $e^{\pm i\hat H_0 t/\hbar}$ sandwich. These oscillating phases will be the seed of the resonance condition: a sinusoidal perturbation at frequency $\omega \approx \omega_{fi}$ drives transitions efficiently because it can cancel the phase $e^{i\omega_{fi}t}$ and let the amplitude $c_f$ build up coherently.

Why the interaction picture is the right tool

Because the free evolution is hidden in the operators, perturbation theory becomes a clean integral. The formal solution of $i\hbar\,\partial_t|\Psi_I\rangle = \hat H'_I|\Psi_I\rangle$ is the Dyson series, whose first term gives the first-order transition amplitude

c_f^{(1)}(t) = -\frac{i}{\hbar}\int_0^t \langle f|\hat H'(t')|i\rangle\, e^{\,i\omega_{fi}t'}\,dt' .

That single integral — derived properly in the next lesson — is the workhorse of the entire module.