这是一份nottingham诺丁汉大学PHYS3001作业代写的成功案例

$$
K_{0}\left(t, \boldsymbol{r}-\boldsymbol{r}^{\prime}\right)=\left(\frac{m}{2 \pi \mathrm{i} t}\right)^{3 / 2} \mathrm{e}^{\mathrm{im} r^{2} /(2 t)}\left[\mathrm{e}^{-\mathrm{i} m \boldsymbol{r} \cdot \boldsymbol{r}^{\prime} / t}+I\left(t, \boldsymbol{r}, \boldsymbol{r}^{\prime}\right)\right]
$$
where
$$
I\left(t, \boldsymbol{r}, \boldsymbol{r}^{\prime}\right)=\mathrm{e}^{-\mathrm{i} m \boldsymbol{r} \cdot \boldsymbol{r}^{\prime} / t}\left[\mathrm{e}^{\mathrm{i} m r^{2} /(2 t)}-1\right] .
$$
Substituting $(9.22 \mathrm{a})$ into $(9.21 \mathrm{a})$, we find that
$$
\chi(t, \boldsymbol{r})=\left(\frac{m}{\mathrm{i} t}\right)^{3 / 2} \mathrm{e}^{\mathrm{i} m r^{2} /(2 t)}[\tilde{\chi}(m \boldsymbol{r} / t)+R(t, \boldsymbol{r})]
$$
where $\tilde{\chi}(m \boldsymbol{r} / t)$ is the Fourier transform of the wave packet $\chi\left(\boldsymbol{r}^{r}\right)$ from (9.21b), i.e.
$$
\tilde{\chi}(m \boldsymbol{r} / t)=(2 \pi)^{-3 / 2} \int \mathrm{d} \boldsymbol{r}^{\prime} \mathrm{e}^{-\mathrm{i} m \boldsymbol{r} \cdot \boldsymbol{r}^{\prime} / t} \chi\left(\boldsymbol{r}^{\prime}\right)
$$
Here, the so-called residual function $R(t, \boldsymbol{r})$ has the form
$$
R(t, \boldsymbol{r})=(2 \pi)^{-3 / 2} \int \mathrm{d} r^{\prime} \mathrm{e}^{-\mathrm{i} m \boldsymbol{r} \cdot \boldsymbol{r}^{\prime} / t}\left[\mathrm{e}^{\mathrm{i} m r^{2} /(2 t)}-1\right] \times\left(\boldsymbol{r}^{\prime}\right)
$$

PHYS3001 COURSE NOTES ：

$$
\left|\left\langle\boldsymbol{r} \mid U_{0}(t) \psi_{0}\right\rangle\right| \leq \frac{C}{t^{3 / 2}} \quad(t>0)
$$
where $C$ is a positive constant. This enables us to write
$$
\left|V U_{0}(t) \psi_{0}\right| \leq \frac{C}{t^{3 / 2}}|V|
$$
which implies that
$$
\begin{gathered}
I\left(t_{0}, \psi_{0}\right)=\int_{t_{0}}^{\infty} \mathrm{d} t|\xi(t)| \leq C|V| \int_{t_{0}}^{\infty} \mathrm{d} t t^{-3 / 2}=2 \frac{C|V|}{t_{0}^{1 / 2}}=\frac{C_{0}|V|}{t_{0}^{1 / 2}} \
I\left(t_{0}, \psi_{0}\right)<\infty \quad\left(t_{0}>0\right)
\end{gathered}
$$
Hence, in the case of short-range potentials $(|V|<\infty)$, the integral $I\left(t_{0}, \psi_{0}\right)$ exists for a certain $t_{0}>0$. Then, the relations $(9.14 \mathrm{~b}, \mathrm{c})$ and $(9.27)$ give
$$
\left|\Omega\left(t_{0}\right) \psi_{0}-\Omega(\infty) \psi_{0}\right| \leq I\left(t_{0}, \psi_{0}\right) \leq \frac{C_{0}|V|}{t_{0}^{1 / 2}}<\infty
$$