Interaction Picture

Why is interaction picture used for TDPT?

The kets can be written in form of time-dependent Schrodinger equation, where $H$ is replace by $V_{I}$ :

i\hbar \frac{\partial}{\partial t} |\alpha, t_0; t\rangle_I = V_I |\alpha, t_0; t\rangle_I \end{align}

The state kets in interaction picture are constant. The time dependence is capture by probability coefficients:

∣ α, t_{0}; t ⟩_{I} = n \sum c_{n} (t) ∣ n ⟩ .

Using the equations above, derive the system of differential equations for the coefficients $c_{n} (t)$ in terms of perturbation $V (t)$ .

i ℏ \frac{d}{d t} C_{n} (t) = m \sum V_{nm} e^{i ω_{mn} t} C_{m} (t), (5.187)

where $ω_{mn} = \frac{( E _{m} - E _{n} )}{ℏ} = - ω_{nm}$ .

Explicitly,

i ℏ \overset{c}{˙}_{1} \overset{c}{˙}_{2} \overset{c}{˙}_{3} ⋮ = V_{11} V_{21} e^{- i ω_{21} t} V_{31} e^{- i ω_{31} t} ⋮ V_{12} e^{i ω_{12} t} V_{22} V_{32} e^{- i ω_{32} t} ⋮ V_{13} e^{i ω_{13} t} V_{23} e^{i ω_{23} t} V_{33} ⋮ c_{1} c_{2} c_{3} ⋮ . (5.189)

Heisenberg, Interaction, and Schrödinger Pictures
	Heisenberg picture	Interaction picture	Schrödinger picture
State ket	No change	Evolution determined by $V_{I}$	Evolution determined by $H$
Observable	Evolution determined by $H$	Evolution determined by $H_{0}$	No change