这是一份liverpool利物浦大学PHYS203/PHYS361的成功案例

The integration over $t$ can be performed by means of the following evident relation:
$$
\frac{\mathrm{d}}{\mathrm{d} t}\left(\mathrm{e}^{\mathrm{i} H t} \mathrm{e}^{-\mathrm{i} H_{0} t}\right)=\mathrm{ie}^{\mathrm{i} H t}\left(H-H_{0}\right) \mathrm{e}^{-\mathrm{i} H_{0} t}=\mathrm{ie}^{\mathrm{i} H t} V \mathrm{e}^{-\mathrm{i} H_{0} t}
$$
which finally yieldswhere $U_{0}(t)$ and $U(t)$ are the evolution operators:
$$
\begin{gathered}
U(t)=\mathrm{e}^{-i H t} \quad(-\infty<t<+\infty) \
U_{0}(t)=\mathrm{e}^{-i H_{0} t} \quad(-\infty<t<+\infty)
\end{gathered}
$$
The domain of the definition of the operators $U(t)$ and $U_{0}(t)$ is the entire Hilbert space $\mathcal{H}$. The family of the operators $U(t)$ satisfies the relation
$$
U(t) U\left(t^{\prime}\right)=U\left(t+t^{\prime}\right)=U\left(t^{\prime}\right) U(t)
$$

PHYS203/PHYS361 COURSE NOTES ：

$$
H_{0}=S^{\dagger} H_{0} S
$$
and, therefore, according to equation (7.32b):
$$
\bar{E}{f}=\bar{E}{i} \quad \text { (QED). }
$$
Here it remains only to provide an explanation for the claim that the quantities $\left\langle\Psi_{i}\left|H_{0}\right| \Psi_{i}\right\rangle$ and $\left\langle\Psi_{f}\left|H_{0}\right| \Psi_{f}\right\rangle$ indeed represent the initial and final energy of the total system. Namely, it is anticipated that the actual total energy $E$ for a certain state $\Psi$ is, in fact, the expected value of the Hamiltonian $H$ :
$$
E=\langle\Psi|H| \Psi\rangle .
$$
However, since we have $\Psi=\Omega \Psi_{0}$, where $\Psi_{0}$ is the eigenfunction of the operator $H_{0}$, it will be
$$
E=\langle\Psi|H| \Psi\rangle=\left\langle\Psi_{0} \mid \Omega^{\dagger} H \Omega \Psi_{0}\right\rangle=\left\langle\Psi_{0}\left|H_{0}\right| \Psi_{0}\right\rangle
$$