Higher Order Perturbation Theory

(This is also discussed in Merzbacher [2] though with a slightly different notation.

We start from the time-dependent Schrödinger equation

$$i\partial_t \vert\Psi(t)\rangle = H(t) \vert\Psi(t)\rangle.$$ (3.16)

The state $\vert\Psi(t)\rangle$ at time $t$ is obtained from the state $\vert\Psi(t_0)\rangle$ at time $t_0$ by application of the time evolution operator $U(t,t_0)$ via

$$\vert\Psi(t)\rangle = U(t,t_0)\vert\Psi(t_0)\rangle.$$ (3.17)

If $H(t)=H$ is time-independent, we have

$$U(t,t_0) = e^{-iH(t-t_0)}, .$$ (3.18)

For arbitrary $H(t)$, we have

$$i\partial_t U(t,t_0) = H(t) U(t,t_0),\quad U(t_0,t_0)=1.$$ (3.19)

We now assume a form

$$H(t) = H_0 + V(t).$$ (3.20)

We solve this differential equation by introducing the interaction picture time-evolution operator $\tilde{U}(t,t_0)$,

$$\tilde{U}(t,t_0) = e^{i{H_0}t} U(t,t_0) e^{-i{H_0}t_0}$$ (3.21)

$$i\partial_t \tilde{U}(t,t_0) = - H_0 \tilde{U}(t,t_0) + i(-i) e^{i{H_0}t} H(t) U(t,t_0)e^{-i{H_0}t_0}$$

$$= - H_0 \tilde{U}(t,t_0) + e^{i{H_0}t} [H_0 + V(t)] e^{-i{H_0}t}e^{i{H_0}t}U(t,t_0)e^{-i{H_0}t_0}$$

$$= - H_0 \tilde{U}(t,t_0) + e^{i{H_0}t} [H_0 + V(t)] e^{-i{H_0}t} \tilde{U}(t,t_0)$$

$$= \tilde{V}(t) \tilde{U}(t,t_0)$$ (3.22)

$$\tilde{V}(t) = e^{i{H_0}t} V(t) e^{-i{H_0}t}.$$ (3.23)