\renewcommand{\ket}[1]{\left\lvert #1 \right\rangle} In typical quantum mechanics courses, we learn about the so-called "Schrodinger picture" and "Heisenberg picture". In the Schrodinger picture, the equation of motion brings time dependence to the states, i \hbar \partial_t \ket{\Psi(t)} = H(t) \ket{\Psi(t)} while in the Heisenberg picture the equation brings time dependence to the operators, i \hbar \partial_t A(t) = [A(t), H(t)] \, .
When we have a Hamiltonian which can be split into an "easy" part H_0(t)^{[a]} and a time dependent "difficult" part V(t), H(t) = H_0(t) + V(t) people talk about the "interaction picture" or "rotating frame". What's the difference between the interaction picture and rotating frame, and how do they work?
[a]: We assume that H_0(t) commutes with itself at different times.
Answer
\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}
Basic idea: the rotating frame "unwinds" part of the evolution of the quantum state so that the remaining part has a simpler time dependence. The interaction picture is a special case of the rotating frame.
Consider a Hamiltonian with a "simple" time independent part H_0, and a time dependent part V(t): H(t) = H_0 + V(t) \, .
Denote the time evolution operator (propagator) of the full Hamiltonian H(t) as U(t,t_0). In other words, the Schrodinger picture state obeys \ket{\Psi(t)} = U(t, t_0) \ket{\Psi(t_0)}.
The time evolution operator from just H_0 is (assuming H_0 is time independent, or at least commutes with itself at different times) U_0(t, t_0) = \exp\left[ -\frac{i}{\hbar} \int_{t_0}^t dt' \, H_0(t') \right] \, . Note that i\hbar \partial_t U_0(t, t_0) = H_0(t) U_0(t, t_0) \, .
Define a new state vector \ket{\Phi(t)} as \ket{\Phi(t)} \equiv R(t) \ket{\Psi(t)} where R(t) is some "rotation operator". Now find the time dependence of \ket{\Phi(t)}: \begin{align} i \hbar \partial_t \ket{\Phi(t)} =& i \hbar \partial_t \left( R(t) \ket{\Psi(t)} \right) \\ =& i \hbar \partial_t R(t) \ket{\Psi(t)} + R(t) i \hbar \partial_t \ket{\Psi(t)} \\ =& i \hbar \dot{R}(t) \ket{\Psi(t)} + R(t) H(t) \ket{\Psi(t)} \\ =& i \hbar \dot{R}(t) R(t)^\dagger \ket{\Phi(t)} + R(t) H(t) R(t)^\dagger \ket{\Phi(t)} \\ =& \left( i \hbar \dot{R}(t) R(t)^\dagger + R(t) H(t) R(t)^\dagger \right) \ket{\Phi(t)} \, . \end{align} Therefore, \ket{\Phi(t)} obeys Schrodinger's equation with a modified Hamiltonian H'(t) defined as H'(t) \equiv i \hbar \dot{R}(t) R(t)^{\dagger} + R(t) H(t) R(t)^\dagger \, . \tag{$\star$} This is the equation of motion in the rotating frame.
Useful choices of R depend on the problem at hand. Choosing R(t) \equiv U_0(t, t_0)^\dagger has the particularly useful property that the first term in (\star) cancels the H_0(t) part of the second term, leaving \begin{align} i \hbar \partial_t \ket{\Phi(t,t_0)} = \left( U_0(t, t_0)^\dagger V(t) U_0(t, t_0) \right)\ket{\Phi(t, t_0)} \, . \end{align} which is Schrodinger's equation with effective Hamiltonian H'(t) \equiv U_0(t)^\dagger V(t) U_0(t) \, . This is called the interaction picture. It is also known by the name Dirac picture.
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